How can Schrödinger's cat be both dead and alive? So, this goes to something so fundamental, I can barely express it. 
The Schrödinger's Cat thought experiment ultimately asserts that, until the box is opened, the cat is both dead AND alive. Now, this is obviously ludicrous. The cat either died or lived at some point; someone opening the box and observing it had zero influence on it.
Saying the cat was both alive and dead till the box was opened seems to be some kind of hardware defect in some people's thinking. I mean, with all respect, I don't know how I can be polite about it.
We humans aren't THAT important. Things happen whether we see them or not. I mean, do I really even need to state that?
The question, then: Is Schrödinger's Cat meant to be taken at all physically?
 A: Remember Heisenberg's idea that you can't always measure position and velocity at the same time? 
So here's an electron, and there's stuff you are guaranteed not to know about it. You can know something about some combination of position and velocity, but that's like having one equation in two unknowns. You know something but you can't solve it like you could with two equations in two unknowns.
Then maybe the electron interacts in some special way. You know its position, and you measure it's velocity. Now you know what it's position and velocity USED TO be, but no longer. For a moment there, you knew. 
Before you measured, you didn't know. You had a probability distribution which gave information you did know about it, but you couldn't know it all. Then you knew. And a moment later you didn't know again but had a new probability distribution.
And Heisenberg says there's no way you could know more.
Here's the point -- we naturally want to think that there is a single reality going on that we can't know about. And there could be. But science is about what we can measure. If there's no possible way to find out about that hidden reality, why should we care about it? If all we know about is probability distributions, why not proceed as if the probability distributions are all that's real?
Logically that works just fine. But people don't like it. But logically it works just fine.
If it's things we can't know about, why choose which way to think about it? If somebody wants to think that invisible undetectable elves are making electrons move the way they do, according to probability functions, why argue with them? Their explanation fits the facts as well as yours does. You could argue that yours is simpler. But so what? Their explanation makes them feel better, and your explanation makes you feel better.
Arguing about explanations for QM which go beyond QM is not physics. It's philosophy. Metaphysics or something. Unless we find a way to find out the things that Heisenberg says we can't find out, it doesn't matter.
But -- Heisenberg doesn't really say you can't know those things. Just that you can't find them out using the things we know about in physics so far. Maybe someday physics will advance to the point that those things do become measurable. 
I think they can't be measured using leptons, hadrons bosons, and the four fundamental forces. (Is it still four forces, or just three, or two? No matter.)
Maybe someday physics will discover new particles and new forces that make it possible. But for now, physics isn't about explanations for QM that can't be measured. That all give the same results.
A: I'll try to answer this without mathematics or sophistry. If it doesn't quite make sense as a result, ask for clarification. This stuff isn't easy to make sense of. So I'm sorry I'd this is a bit of a long answer!
Schroedinger's cat is a way to visualise something that we usually only observe at subatomic level, in a more everyday context. That's actually quite important:  when you imagine it at an everyday scale, it is easier to see the point Schroedinger was making - but its a lot harder to see how any of it makes sense. 
So I'm going to roll it back to subatomic level a bit.
Uncertainty in the universe
Schroedinger was facing the question of how to interpret quantum mechanics equations. The equations seemed to suggest that reality as we know it, can be described by an equation that is a bit like equations used to model waves and other phenomena that develop and change over time.
But waves also do other things, they can superimpose:  for example, two waves can pass through each other temporarily creating a larger wave. Does this suggest that reality can also superimpose? 
Also, Heisenberg had already proposed that when you looked closely at reality, it wasn't deterministic - there was uncertainty in it. You couldn't measure all things accurately, and if you tried to measure some, you'd alter other measurements when you tried.  The laws of nature themselves didn't let you measure everything with precision.
These effects can be seen on a subatomic level - a huge part of modern science uses them for everyday devices like lasers and computers. But the equations say they also apply to all things in the universe, big and small, its just that it is much harder to see them on a big object scale. But sometimes, we do see them, even on a big scale.  Superconductivity, superfluidity, lasers, quantum tunnelling based devices, ...  these all work. We can observe them. 
So we have a good idea these equations are valid.  We are pretty sure that probability and uncertainty are fundamental to how physical reality "works".   But what do these equations actually mean, and what are they saying about physical reality?
The cat and the particle
Schroedinger's cat is a way to imagine these uncertainties, scaled up. It imagines we tie the fate of a cat, to the fate of an unstable radioactive subatomic particle. These particles break up, but you cant predict when they'll break up. You can only say they have such and such a chance of breaking up in an hour, or a year, or a billion years.  If the particle breaks up (radioactive decay), the cat dies. If the particle doesn't break up, the cat lives. So thinking about the cat, gives us a way to ask questions about the particle's breaking up, and what state the particle is in.
It's actually a bit easier to think about the subatomic particle than the cat. So I'm going to do that.
The equations of quantum mechanics don't tell us with certainty that the particle has or hasn't broken up. Instead, they tell us that at this point in time, the particle has (say) a 40% chance of having broken up and a 60% chance of not broken up.... and that nothing you can do, will tell you which it is, until you actually look at it ("observe" it).
So Schroedinger asks this question:   According to these equations, what is the state of the particle when we haven't yet looked at it, and then when we do look? The answer is hard to understand, and the current best understanding we have, is like this:

Before we look, the particle is in some state (intact or broken up), but we can't know which.   It effectively behaves as if its a combination made up of 40% of a broken up state, and 60% of an intact state. When we observe it, we find which one it really is, now, and then it acts like it's 100% that state.
That change from 60%/40% to 100% is triggered by an observation. That change is what scientists mean when we say an observation "collapses" the states.
Before observing there were two possible states with 60% and 40% likelihoods, kind of overlaid over each other ("superimposed"). We literally could not know which was "the actual state", and mathematically it seems to behave as if it was 60% one and 40% the other, until we "observe" it.  When we do eventually observe it, we see one state, not two. But we can't predict which of them it will be. We can only say that its 60% likely to be one, and 40% likely to be the other.
We don't fully understand what makes an "observation" special or have that effect, in a real world sense, but it seems that's how it is. 

That's really hard to understand, so Schroedinger describes it by analogy, using a cat instead. But I think its sometimes easier to visualise by thinking about the subatomic particle he's really talking about.
A: Before reading this answer (and to those who are downvoting), I am addressing if the cat is both alive and dead. I don't think the question is asking for a complete explanation of the Schrodinger's cat experiment, nor is it asking how this links to all of the deeper mysteries of quantum mechanics and how we should think of them. Therefore, while there is much to be gained in thinking of many different interpretations, I will not be addressing them here.

Schrodinger's cat is not both dead and alive any more than an electron simultaneously exists at every point in space. You are using a pop-sci explanation of Schrodinger's cat that indeed falls apart when you dig deeper.$^*$ The key point is that a system cannot be in multiple states at once.
Schrodinger's cat (or if you hate this example, think "quantum system") is always in a single state. Typically the example says that there is an equal probability of us "measuring" the cat to be either alive or dead once we open the box. Therefore, the cat is in a state that is a superposition of our "life states" $|\text{alive}\rangle$ and $|\text{dead}\rangle$:
$$|\text{cat}\rangle=\frac{1}{\sqrt{2}}\left(|\text{alive}\rangle+|\text{dead}\rangle\right)$$
This state tells us that there is a probability of $0.5$ of observing the cat as alive and a probability of $0.5$ of observing the cat as dead. This is because 
$$|\langle\text{alive}|\text{cat}\rangle|^2=0.5$$
$$|\langle\text{dead}|\text{cat}\rangle|^2=0.5$$
Once we open the box (perform a "life state" measurement of the system), the state of the cat collapses to one of the life states (eigenstates of the "life measurement operator"). So we observe the cat as either alive or dead.
It is important to understand that before we open the box the cat is not both alive and dead. The system cannot be in multiple states at once. It is in a single state, and this state is described as a superposition of life states. Once we open the box the cat is in a new single state which is one of the two life states. We cannot determine which state the cat ends up in though, only the probabilities it will end up in a certain state.

Of course Schrodinger's cat is crazy to think about because we are trying to apply QM formalism to the macroscopic world, but this is precisely how quantum systems work. We can express the state $|\psi\rangle$ of a quantum system as a superposition of eigenstates $|a_i\rangle$ of a Hermitian operator $A$:
$$|\psi\rangle=\sum_ic_i|a_i\rangle$$
We do not say that the system is in every state $|a_i\rangle$ at once. It is in a single state (the superposition) that tells us the probability $|c_i|^2$ of the system being in one of the states $|a_i\rangle$ after making a measurement of the physical quantity associated with operator $A$.

$^*$I will use the Copenhagen interpretation of QM for my answer, since it is the most widely used interpretation to teach introductory QM. This is just one way to view this thought experiment, and it certainly is not a complete explanation. There are other interpretations that get to deeper meanings, more practical understand of measurements, etc. For that I'll refer you to the other answers, but I am not claiming this is the only way to view this scenario or QM in general. This question is not asking for a full explanation of the Schrodinger's cat experiment with a look into the deeper meaning of QM, so I am not going to get into all of that. The main point of this answer does not depend on the QM interpretation anyway.
A: Basically the answer is yes, the cat is both dead and alive. People used to discuss this sort of thing in terms of the Copenhagen interpretation (CI) and the Many-Worlds interpretation (MWI), but those discussions tend not to be satisfying, because both CI and MWI are designed so that in almost all real-world measurements, they give the same predictions. A better way to talk about this is in terms of decoherence.
Quantum mechanics says that the cat is in a superposition of states, alive and dead. Quantum mechanics doesn't impose any maximum size on objects that can be in a superposition of states. Double-slit interference has been observed with large molecules https://arxiv.org/abs/1310.8343 , and there are serious proposals to do it with a virus:  http://arxiv.org/abs/0909.1469
However, due to interaction with its environment (e.g., vibrations from the walls of the box, and infrared radiation), the definite phase relationship between the live and dead parts of the cat's wavefunction would be lost very rapidly -- the time scale for a cat in a box would be many orders of magnitude too short to allow us to do anything during that time. Once the phase information is effectively lost, it becomes impossible to observe wave interference effects between the live and dead cat.

We humans aren't THAT important. Things happen whether we see them or not. 

Right, this was always one of the unsatisfactory things about CI. Decoherence actually happens regardless of whether we observe the object at all. Our interaction with the system would cause decoherence, but so do other interactions, and they do it on much shorter time scales.

I can only consider it a fundamental breakdown of seemingly intelligent minds.

Lots of things in physics are counterintuitive.
A: It isn't - or, better, a cogent understanding of quantum theory says we cannot completely know in this case, what is going on inside the box, without opening it up.
I have long maintained for a while that the crucial distinction between "classical" and "modern" physics is not so much something like "absolute space and time" vs. "relative and 'gooey' space and time", or "determinism" vs. "indeterminism" or anything else like that, but rather that it is that modern physics is physics in which specific, and fundamental (not derivable from anything else) new laws come into play that specially regard the behavior of information in the Universe, and I think a lot of the inability and misunderstandings surrounding modern physics generally comes from a sad historical accident in which that information theory was discovered later than the groundwork for such was first laid.
In the case of quantum theory, this manifests itself in the following. The most cogent understanding, I have found, is that the chief object of concern, the "mysterious" quantum state vector, is something that should not be attributed as being, or at least it cannot by default be identified with, a "property of the system". This is really a hang-over from pre-modern, Newtonian viewpoints, and the whole modern physics programme has been the progressive demolition of these as being the most useful when it comes to creating highly accurate ways of describing physical phenomena. Rather, what it is it is a mathematical model - so even in this understanding, we should not take it too "literally" - of information which is held by an "agent" about the system, in this case, the "cat".
That is, the $|\psi\rangle$ thingy that most commonly appears in these discussions does not, strictly speaking, belong "to the cat". It "belongs" to you, or, well, a mathematical model of "you" or something else suitable in "your" place. And that's an important point to make because that is necessary to disabuse one of the notion that quantum theory necessarily must be "anthrocentric": "agents" in this context really are just systems which are capable of storing information, acquiring information from the outside world by an interaction therewith, and then updating their internal information stores based on such input. Another perfectly good agent could be a non-human animal (yes, the cat), another could be a computer with sensors and recording data on a storage drive, another could be a robot of some sort with a reactive control system - as long as it meets the above properties, it's an agent. That's it. 
What $|\psi\rangle$ describes regarding "you" is what "you" - the fictive one in the story that we are telling with the theory - what information you possess, and what it signifies. There are two basic terms, in this setup which are
$$|\mathrm{alive}\rangle$$
and
$$|\mathrm{dead}\rangle$$
which don't necessarily correspond to "the cat is alive" and "the cat is dead" but rather "your information describes a cat being alive" and "your information describes a cat being dead". Note that I have chosen that phrasing very carefully, and that's very important, because the difficulties in all this that often come up really boil down to not being careful and subtly equivocating or reifying things in ways that end up violating whatever assumptions one tries to make at the outset. I did not say "the cat is" something, and it is extremely important to wean one from that idea. There is no supposition that the information pertains "accurately" to the cat, rather it is information solely in a "register" in "your" (or "the agent's") "mind" (or its hard disk, or RAM sticks, or whatever). The information here pertains to the question "Is the referent cat alive or dead?" and $|\psi\rangle$ encodes an answer to that question, stored by the agent. You should perhaps think of it less as a state and more as a datum , a piece of data, and hence I will refer to it as a datum from here on to reinforce this notion.
The "weird" data of the form
$$|\psi\rangle_\mathrm{weird} := \alpha |\mathrm{alive}\rangle + \delta |\mathrm{dead}\rangle$$
in this understanding, which cause all the trouble, are really just ways of saying, (after normalization so that $|\alpha|^2 + |\delta|^2 = 1$) "My information describes a cat being alive with probability $|\alpha|^2$ and dead with probability $|\delta|^2$". This information is incomplete: it doesn't say "the cat is simultaneously alive and dead", it says that the yes/no answer is less definite as to which one. This can be made a bit more rigorous thanks to the use of Shannon entropy, which basically measures, given an "answer" to a question stated as a probability distribution, how much it is lacking in information:
$$H[X] := -\sum_i P_i \lg P_i$$
which in this case becomes
$$H_\mathrm{aliveness}[|\psi\rangle_\mathrm{weird}] = -(|\alpha|^2 \lg |\alpha|^2 + |\beta|^2 \lg |\beta|^2)$$
and it allows us to say that the datum gives an answer to the question that is "worth" between 0 and 1 bit, i.e. a fraction of a bit, instead of being always worth 1 bit, as if I said "yes, it's alive" or "no, it isn't".
So now what? Well, in modeling the situation with quantum theory, being very careful, it goes like this. We (cruelly) put the (living) cat in the box and close it tight. Our current knowledge is summarized as
$$|\mathrm{alive}\rangle$$
We wait an interval $t_f$. After that time, we open the box. Our knowledge then becomes either
$$|\mathrm{alive}\rangle$$
(i.e. no change)
or we saw something else, i.e.
$$|\mathrm{dead}\rangle$$
. That's it.
That's the whole "kit 'n kaboodle" when it comes to this. In these two cases, and only these two cases, is it "safe" to say that the information in question "corresponds with" "reality", in that the "real" answer as to the "aliveness" of the cat is one of these. When we aren't looking, we cannot say. "Looking" in the box is the interaction through which we acquire the new information.
What we can do, when we are not looking, is to use the Schroedinger equation - and the role of this should not be unduly elevated: what it really is is a predictive device, or an inference rule, that lets us deduce from a datum of information regarding a present situation, what it implies informationally about a future situation. When we do this with the cat, starting initially with $|\mathrm{alive}\rangle$, we get data that are a blend of $|\mathrm{alive}\rangle$ and $|\mathrm{dead}\rangle$ in the "weird" form above. But note: this is not, under this framework, necessarily what is "really" there. We can't know - it is just a prediction, and it gives a probability balance. If you want to think of it as anything, you should think of it likeyour weather forecaster, who can only give you the most probable result for what the weather will be. Hence, when it says what sseems like "the cat is 50% 'alive' and 50% 'dead'", it doesn't mean that under this understanding: it means "the predicted information is that at this time t, we can assign a 50% probability that if we query it now, we will get 'alive' and a 50% probability that if we query it now, we will get 'dead'".
Insofar as the "reality" is concerned - there are many different possible things that could happen here that could result in that coming out to be the case: and that's the thing, the theory doesn't provide enough information to tell which is which. That's why, as I said, it's a subjective theory.
But this is then where many others will get hung up - they tend to think that if it's a subjective theory, it must then "cloak" some kind of reality where the "real" answers are, in fact, "binary" in the sense that the "truth", no matter how it operates, is that a parameter of the system must have a value like "alive" or "dead" or "5.000... joules" or something like that at all times and then noting how this runs into various sorts of trouble.
But I would challenge that thusly: Quantum theory, under the assumptions we just gave - that the datum vector is subjective information, that upon querying / "measuring" a question or parameter it then acquires at that time "real" information, and the assumption of relativistic causality - suggests there are situations where in some sense the "real" parameters actually are these things we would define with probability distributions - in some sense, the "fraction of a bit" with "75% likely to be A and 25% likely to be B" is "all that exists" for that parameter, "for real".
And the way to see this is to consider systems more complicated than Schrodinger's cat: such as the electron in a hydrogen atom. If one performs what is called a "full-determination measurement", a query that acquires the values of the three electron quantum numbers $n$, $l$, and $m$, then at that point, the position and momentum parameters are fuzzy, i.e. they have probability distributions and nontrivial Shannon entropy, the quantum theory doesn't allow us to, say, add additional information (decrease the Shannon entropy) without then also losing information in the quantum numbers. Taking that such a measurement gives "as much information as possible", we have no really good choice except to conclude that this is the "real" amount of information that "really exists" and, hence, also when we aren't looking, it could also be that parameters, such as the cat's "alive/dead" state, are also similarly limited, but we don't know, because that's the limit of the theory's "objectivity".
If it does, though, it's not right to say the cat is both dead and alive, so much as to say that if the state referred to as such obtains "for real" in between, it is in some state in which the parameter represented by the "alive/dead" question has a fraction of an answer in it - a fractional bit, as Shannon would describe.
A: I feel like all the answers here are missing the point.
The cat is not both alive and dead at the same time. That would be, as you put it, ludicrous. The truth is that the cat is in a superposition state of the states "alive" and "dead".
The problem is that there is no way to make sense of this statement without studying the underlying mathematics. Humans have no intuition for the concept "superposition", but some very smart people have found out that this concept describes our reality.
When scientists are asked to describe the experiment in layman's terms, they cannot say "you have to study the underlying mathematics". So they make their best effort to appeal to the layman's intuition by saying that the cat is both alive and dead at the same time. This is of course wrong, but there is simply no better way to phrase it in layman's terms.
A: Forgive the length.  I find Schrödinger's cat is much easier to make sense of as a journey through QM, rather than just a few equations that someone says "solves your problems."
Schrödinger's cat was definitely meant to be taken seriously, in that it was intended to be a serious challenge to naively applying the Copenhagen interpretation to macroscopic objects.
The general challenge brought forth is that constructions like Schrödinger's cat have so many particles, thus an enormous state space, such that simplifying it down into binary states like "alive" and "dead" yields incorrect results.
The real trick to the experiment is the element which is oft overlooked.  It's not the cat, or the radioactive isotope.  It's not even the box.  It's the detector inside the box.  You question whether we are "special" enough to collapse a waveform.  It's actually not us opening the box that will cause a collapse, but the detector.  It's job is to take a quantum level event of "a particle that has a 50% chance of decaying during the experiment" into "a classical measurement of whether the particle decayed," which we then use to signal the machine to use the hammer to smash the vial of poison  Just putting such a detector in a box doesn't make it any less of a detector.  It's still doing the classical thing.
So what if we wanted to treat the detector as a quantum thing?  After all the point of Schrödinger's cat is to poke and prod at what happens if we try this?
Well now we have to be a bit more careful.  We have to consider not only the state of the cat and the isotope but also the state of the detector.  And the detector seems to be the tricky bit, as it's job is to go quantum to classical, and that makes it interesting.
So what's such a big deal about a quantum thing anyways?  Why do we need to have such a confusing model of the world.  For the most part (read: everything you or I will experience in our lives unless we become a physicist or some flavors of engineer) is well described with "classical" behaviors.  These don't confuse us.  However, there are some situations which arise at atomic scales which simply act "odd."  We find situations where particles appear to teleport through walls or take two paths at the same time.  To make sense of those, we needed new math.
The new rules are, statistically speaking, a superset of the old ones.  In most situations, we have lots and lots of particles.  We don't know their state, but we can know probabilistic what their state distributions look like.  If you run these new rules over large sets of particles for long periods of time, you get the same results you expected from classical thinking (okay, maybe "long by quantum standards"  milliseconds are a long time for many quantum systems!)
More to the point of Schrödinger's cat, these new rules obey a principle known as "superposition."  In Aaron Steven's answer, he was very careful to point out that the cat exists in exactly one state at all times.  There's a good reason he was so careful there.  When we write something like $|\text{cat}_{initial}\rangle=|\text{alive}\rangle$ or $|\text{cat}_{final}\rangle=\frac{1}{\sqrt{2}}\left(|\text{alive}\rangle+|\text{dead}\rangle\right)$, we are describing the one and only state that the cat is in.  However, by the rules of superposition (which all quantum systems obey), we can figure out the state the cat will be in by looking at each branch of an addition, one at a time, and then add them up later (Formally, we can say that for any linear operation $f$ on the system $(f(x_1+x_2) = f(x_1) + f(x_2)$).  This is convenient for you and I, because we are much more comfortable thinking through what happens to an "alive" cat or a "dead" cat, rather than trying to handle some complex mathematical equations that links both.  The fact that QM wavefunctions have this superposition property lets us do this rigorously.*
And, indeed, for observations, we arrive at the same thing Aaron described.  The probability of us observing the cat as alive is 50%.  It behaves precisely as if the alive/dead variable was merely unknown until we open the box.  There are no surprises there.
But the story isn't done, because there's other things we can do to the box.
There are operations we can do which don't operate in such simple ways as our classical observations do.  Quantum operators are fascinating linear functions which can do things we don't always expect.  After all, that's why we have QM.  And this is why the sensor matters.
We can operate on the cat/box/sensor/particle system with a quantum operator if we like.  And, if I may be a bit informal with it, the system after interaction might be $|\text{cat}_{after}\rangle=a|\text{alive}\rangle+b|\text{dead}\rangle+c|\text{weird}\rangle$, where $a$ $b$ and $c$ are just real numbers.  The $|\text{alive}\rangle$ handles the cases which are handled intuitively as having an alive cat, $|\text{dead}\rangle$ handles the cases which are handled intuitively as having a dead cat, and $|\text{weird}\rangle$ lumps together all of the really wonky cases where quantum mechanics says one thing where our intuition says another.  One of the great things about the bra-ket notation that physicists like to use is I can use it to correctly capture a system, even when using really oddball states like "weird."
So now we come back to the detector.  This detector could have been any system really.  There's more interesting things to throw into a box with a cat, but the experiment calls for a detector.  And, hand-waving emphatically, one aspect of a good detector in physics land is that it minimizes the probability of any weird things happening.  Using the above equation, we try to design sensors in such a way that, for any interaction one may wish to do with the system (opening the box, or any quantum operator), the constant $c$ in $c|\text{weird}\rangle$ is vanishingly small ($c\approx 0$).  A sensor which doesn't have this property is a pretty poor sensor, and I would no longer be comfortable with the intuitive idea that it "detects" the radioactive isotope decaying.
So this detector (which itself has a macroscopic state) was designed to make it incredibly hard to operate on the system in any way which distinguishes it from the simple alive or dead cases which were well described by being "unknown" earlier.  Its job is to make the whole "collapse when you open the box" idea defunct, because the observation already happened inside the box by the detector.
Now you can construct more interesting experiments with things other than nice clean detectors.  And you can start to see quantum effects at the macroscopic level.  There's an entire approach to QM around studying "decoherence" which handles this in a statistically rigorous way and does a good job predicting the results of more odd systems that permit more $|\text{weird}\rangle$ through by design.  For example, there's a whole approach of using "weak measurements" which are measurements designed to not disturb "weirdness" that was already happening in the experiment.  But in this case we can comfortably say the detector "collapsed" the wave form.  And, approaching the topic through the idea of decoherence, we can even show why that term is valid: we intentionally designed the detector to "collapse" the weird part of the waveform into a vanishingly small part.
So never forget the detector.  It was a small part of the experiment, but it turns out to be where the devil decided to put all his details.
*. As a perhaps useful aside, the decomposition itself isn't all that important.  This could have been $|\text{cat}\rangle=a|\text{male}\rangle+b|female\rangle$, describing what happened to the cat if it was male or the cat if it was female.  The math would actually end up right either way.  However, by selecting states which are convenient to the human doing the math (alive and dead), it becomes easier to leverage the superposition principle to actually start picking away at the problem, rather than merely developing new bases.
A: Schrodinger's cat is an exercise in seeing how nonsensical the Copenhagen interpretation is, so answers that attempt to clarify it in terms of CI are not very helpful, in my opinion.
As a framework for this answer, I'll repeat a point I make frequently: QM describes not the probabilistic evolution of a single deterministic state, but rather the deterministic evolution of a probability model of possible observed states. Like any probability model, you can have conditional probabilities (the probability of one event conditioned on another). While, for mechanical computations you might use CI to envision QM as starting with a deterministic state and evolving it into superpositions, in reality it's a probability model all the way along, and any observed initial condition is not an initial state but rather an event in the probability model on which the probabilities of other events can be conditioned.
In the setup of Schrodinger's cat, you have all sorts of observations which are events in the probability model, such as the existence of something that looks like a cat, which of course is an aggregate of lots of smaller events. The events of finding a live cat in the box and the event of finding a dead cat in the box both have probabilities within the probability model that don't change regardless of what happens in the experiment, even after you see the result, because the model encompasses all possibilities (after all, it's compatible with MWI, regardless of whether you assign any ontological significance to MWI). Of course their conditional probabilities conditioned on other observed events will be different and will depend on those.
All of this makes the most sense in terms of a (necessarily non-local, thanks to Bell's Theorem) hidden variable interpretation of QM where the hidden variable is just which of MWI's many worlds is actually the "real" one you're living in.
A: 
How can Schrödinger's cat be both dead and alive?

It is necessary to say once more that Schrodinger's cat is a bad example of quantum mechanical concepts.
What is the quantum mechanical experiment? The decay of a particle which, detected by a detector triggers a poison. The cat is inhumanely used as a magnifier of that trigger. If instead of the series: poison-cat one had a recording of a geiger counter intercepting the decay (instead of poison) one, by accumulating many counts, would have the measurement of the decay curve in time  of that particle and then could identify which type the radioactive sample was. All the rest is philosophical miss direction.
The experiment is one hit in a probability accumulation versus time for measuring the decay curve of a given particle.
The question "is the cat alive or dead" is the same as asking "has this specific atom decayed"? This can only be answered by a statistical accumulation, and it has no meaning by itself other the being a throw of the dice for the decay of that type of particle.
It is a bad example because instead of stressing the novelty of the quantum mechanical frame to depend on probability amplitudes, makes a conundrum that cannot be answered at the macroscopic level of cat and box. Take this example:
Mike and Chris are two friends who lost contact, one is in the UK and the other in the US. The probability of being hit by a car  and dying is X . Mike should think according to the Schrodinger cat, that Chris is both alive and dead.
My answer here is relevant to the matter of decays.
A: I think there are some facts you should know before understanding this answer. Einstein and Schrödinger used the thought experiment to "explain" a point that was wrong. They thought it was absurd for quantum mechanics to say that the state a|alive⟩+b|dead⟩ was possible in Nature (it was claimed to be possible in quantum mechanics) because it allowed the both "incompatible" types of the cat to exist simultaneously.
They were wrong because quantum mechanics does imply that such superpositions are totally allowed, must be allowed, and this fact can be experimentally verified – not really with cats but with objects of a characteristic size that has been increasing. Macroscopic objects have already been put to similar "general superposition states".
The men introduced it to fight against the conventional, Copenhagen-like interpretations of quantum mechanics, and that's how most people are using the meme today, too. But the men were wrong, so from a scientifically valid viewpoint, the thought experiment shows that superpositions are indeed always allowed – it is a postulate of quantum mechanics – even if such states are counterintuitive. Similar superpositions of common-sense states are measured so that only $|a|^2$ and $|b|^2$ from the coefficients matter and may be interpreted as (more or less classical) probabilities. Due to decoherence, the relative phase is virtually immeasurable for large, chaotic systems like cats, but in principle, even the relative phase matters.
A: The cat is not dead or alive at the same time. This is because the cat is not an isolated quantum system that doesn't interact with the universe (experts say, the decoherence time of the cat is extremely small of the order of $10^{-40}$ s, see e.g. https://arxiv.org/pdf/quant-ph/0306072.pdf p.14 ). But your knowledge about the state of the cat is incomplete. So as long as you don't open the box, your knowledge about the cat is only that it is either dead or alive.
Actually in the end this is just statistical reasoning. For example as long as you haven't thrown a dice your knowledge about it is a "superposition" of the states 1,2,3,4,5,6. 
Update: The more I think about it it seems to me that this thought experiment tells us more about the box than about the cat. Because the real question is: Is it possible to make a box around a macroscopically object, which prevents any interaction with the universe, so the cat stays in a superposition and doesn't decohere. I believe the only classical thing that could resemble such a box would be a black hole. But if you put a cat and poison into a black hole, you cannot "open" the box anymore to see the outcome of the experiment.
A: Short answer: the cat isn't dead and alive. It's dead or alive. 
When faced with a quantum superposition of more than one state, e.g.,
$$
|\phi\rangle = \frac{1}{\sqrt{2}}\left(|a\rangle + |b\rangle\right),
$$
interpret the plus to mean that when measured the state could be found to be $a$ or $b$, with probabilities governed by the Born rule (i.e., 1/2 in this case). 
