What are the most fundamental assumptions of quantum mechanics? Can anybody give a list of the most fundamental assumptions of quantum mechanics in plain english ?
 A: Here are some ways of looking at it for an absolute beginner.   It is only a start.  It is not the way you will finally look at it, but there is nothing actually incorrect here.
First, plain English just won't do, you are going to need some math.  Do you know anything about complex numbers?   If not, learn a little more before you begin.
Second, in quantum mechanics, we don't work with probabilities, you know, numbers between 0 and 1 that represent the likelihood of something happening.   Instead we work with the square root of probabilities, called "probability amplitudes", which are complex numbers.   When we are done, we convert these back to probabilities.   This is a fundamental change; in a way the complex number amplitudes are just as important as the probabilities.  The logic is therefore different from what you are used to.
Third, in quantum mechanics, we don't imagine that we know all about objects in between measurements, rather we know that we don't.   Therefore we only focus on what we can actually measure, and possible results of a particular measurement.   This is because some of the assumed imagined properties of objects in between measurements contradict one another, but we can get a theory where we can always work with the measurements themselves in a consistent way.    Any measurement (meter reading, etc.) or property in between measurements has only a probability amplitude of yielding a given result, and the property only co-arises with the measurement itself.
Fourth, if we have several mutually exclusive possibilities, the probability amplitude of either one or the other happening is just the sum of the probability amplitudes of each happening.    Note that this does not mean that the probabilities themselves add, the way we are used to classically.   This is new and different.
If we add up the probabilities (not the amplitudes this time but their "squares") of all possible mutually exclusive measurements, gotten by "squaring" the probability amplitudes of each mutually exclusive possibilities, we always get 1 as usual.   This is called unitarity.
This is not the whole picture, but it should help somewhat.
As a reference, try looking at three videos of some lectures that Feynman once gave at Cornell.   That should help more.
A: I don't really understand exactly what you mean with assumptions, but I guess you ask about the physical nature and behavior of the quantistic world. I am going to be very inaccurate in the following writing, but I need some space to get things understood. What you are going to read is a harsh simplification.
The first assumption is the fact that particles at the quantistic level don't have a position, like real-life objects, they don't even have a speed. Well, they sort of have it, but in reality they are "fuzzy". Fuzzy in the sense that you cannot really assign a position to them, or a speed. 
To be fair, you perfectly know a real-world case scenario where you have to give up the concepts of speed and position for an entity. Take a guitar, and pluck a string. Where is the "vibration" ? Can you point at one specific point in the string where the "vibration" is, or how fast this vibration is traveling. I guess you would say that the vibration is "on the string" or "in the space between one end of the string, and the other hand". However, there are parts of the string that are vibrating more, and parts that are not vibrating at all, so you could rightfully say that a larger part of the "vibration" is in the center of the string, a smaller part is on the lateral parts close to the ends, and no vibration is at the very ends. In some sense, you will experience higher "presence" of the vibration in the center.
With electrons, it's exactly the same. Think "electron" as "vibration" in the previous example. Soon, you will realize that you cannot really claim anything about the position of an electron, for the fact that has wave nature. You are forced to see the electron not as a charged ball rolling here and there, but as a diffuse entity totally similar to the concept of "vibration". As a result, you cannot say anything about the position of an electron, but you can claim that there are zones in the space where the electron has "higher presence" and zones that have "smaller presence". These zones are probabilistic in nature, and this probability is directly obtained by a mathematical description called the wavefunction. 
The wavefunction mainly depends on the external potential, meaning that the presence of charges, such as protons and other electrons, will affect the perceived potential and will have an effect on the probability distribution in space of an electron. 
A second assumption is the mapping between physical quantities (such as energy, or momentum) and quantistic operators. Take for example the momentum of an everyday object: it is given by its mass times its speed. In the quantistic world, you don't really have the concept of position, hence you don't have the concept of speed, hence you have to reinterpret the whole thing in terms of probability (rememeber the string ?), which means that what you have about a quantistic object is its wavefunction, and you have to extract the information about the momentum from this wavefunction. How do you do it ?
Well, there's a dogma in quantum mechanics, that if you apply a magic "operator" to the wavefunction, it gives you the quantity you want. There are (simple) rules to generate these operators, and you can apply them to a wavefunction to query the momentum, or the total energy, the kinetic energy and so on.
A: In a rather concise manner, Shankar describes four postulates of nonrelativistic quantum mechanics. 
I. The state of the particle is represented by a vector $|\Psi(t)\rangle$ in a Hilbert space.
II. The independent variables $x$ and $p$ of classical mechanics are represented by Hermitian operators $X$ and $P$ with the following matrix elements in the eigenbasis of $X$
$$\langle x|X|x'\rangle = x \delta(x-x')$$
$$\langle x|P|x' \rangle = -i\hbar \delta^{'}(x-x')$$
The operators corresponding to dependent variable $\omega(x,p)$ are given Hermitian operators
$$\Omega(X,P)=\omega(x\rightarrow X,p \rightarrow P)$$
III. If the particle is in a state $|\Psi\rangle$, measurement of the variable (corresponding to) $\Omega$ will yield one of the eigenvalues $\omega$ with probability $P(\omega)\propto |\langle \omega|\Psi \rangle|^{2}$. The state of the system will change from $|\Psi \rangle$ to $|\omega \rangle$ as a result of the measurement. 
IV. The state vector $|\Psi(t) \rangle$ obeys the Schrödinger equation
$$i\hbar \frac{d}{dt}|\Psi(t)\rangle=H|\Psi(t)\rangle$$
where $H(X,P)=\mathscr{H}(x\rightarrow X, p\rightarrow P)$ is the quantum Hamiltonian operator and $\mathscr{H}$ is the Hamiltonian for the corresponding classical problem.
After that, Shankar discusses the postulates and the differences between quantum mechanics and classical mechanics, hopefully in plain english. Probably you want to take a look at that book: Principles of Quantum Mechanics
There are, of course, other sets of equivalent postulates.  
A: Dirac seems to have thought that the most fundamental assumption of Quantum Mechanics is the Principle of Superposition  I read somewhere, perhaps an obituary, that he began every year his lecture course on QM by taking a piece of chalk, placing it on the table, and saying, now I paraphrase:
One possible state of the piece of chalk is that it is here.  Another possible state is that it is over there (pointing to another table).  Now according to Quantum Mechanics, there is another possible state of the chalk in which it is partly here, and partly there.
In his book he explains more about what this « partly » means: it means that the properties of a piece of chalk that is partly in state 1 (here) and partly in state 2 (there), are in between the properties of state 1 and state 2.  
EDIT: I find I have compiled the basic explanations given by Dirac in the first edition of his book.  Unfortunately, it is relativistic, but I am not going to re-type it all.  
« we must consider the photons as being controlled by waves, in some way which cannot be understood from the point of view of ordinary mechanics.  This intimate connexion between waves and particles is of very great generality in the new quantum mechanics. ...
« The waves and particles should be regarded as two abstractions which are useful for describing the same physical reality.  One must not picture this reality as containing both the waves and particles together and try to construct a mechanism, acting according to classical laws, which shall correctly describe their connexion and account for the motion of the particles.
« Corresponding to the case of the photon, which we say is in a given state of polarizationn when it has been passed through suitable polarizing apparatus, we say that any atomic system is in a given state when it has been prepared in a given way, which may be repeated arbitrarily at will.  The method of preparation may then be taken as the specification of the state.  The state of a system in the general case then includes any information that may be known about its position in space from the way in which it was prepared, as well as any information about its internal condition.
« We must now imagine the states of any system to be related in such a way that whenever the system is definitely in one state, we can equally well consider it as being partly in each of two or more other states.  The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannnot be conceived on classical ideas.
...
« When a state is formed by the superposition of two other states, it will have properties that are in a certain way intermediate between those of the two original states and that approach more or less closely to those of either of them according to the greater or less `weight' attached to this state in the superposition process.
« We must regard the state of a system as referring to its condition throughout an indefinite period of time and not to its condition at a particular time, ... A system, when once prepared in a given state, remains in that state so long as it remains undisturbed.....It is sometimes purely a matter of convenience whether we are to regard a system as being disturbed by a certain outside influence, so that its state gets changed, or whether we are to regard the outside influence as forming a part of and coming in the definition of the system, so that with the inclusion of the effects of this influence it is still merely running through its course in one particular state. There are, however, two cases when we are in general obliged to consider the disturbance as causing a change in state of the system, namely, when the disturbance is an observation and when it consists in preparing the system so as to be in a given state.
« With the new space-time meaning of a state we need a corresponding space-time meaning of an observation. [, Dirac has not actually mentioned anything about `observation' up to this point...]
A: Uncertain Principles: 7 essential elements of QM
Sorry but I can't be made plainer than that in English. The link has been provided by @user26143 , the copy by @DeerHunter . 

Posted on: January 20, 2010 11:13 AM, by Chad Orzel  ( who happens to be a USER here, on Physics.SE).
1) Particles are waves, and vice versa. Quantum physics tells us that every object in the universe has both particle-like and wave-like properties. It's not that everything is really waves, and just sometimes looks like particles, or that everything is made of particles that sometimes fool us into thinking they're waves. Every object in the universe is a new kind of object-- call it a "quantum particle" that has some characteristics of both particles and waves, but isn't really either.
Quantum particles behave like particles, in that they are discrete and (in principle) countable. Matter and energy come in discrete chunks, and whether you're trying to locate an atom or detect a photon of light, you will find it in one place, and one place only.
Quantum particles also behave like waves, in that they show effects like diffraction and interference. If you send a beam of electrons or a beam of photons through a narrow slit, they will spread out on the far side. If you send the beam at two closely spaced slits, they will produce a pattern of alternating bright and dark spots on the far side of the slits, as if they were water waves passing through both slits at once and interfering on the other side. This is true even though each individual particle is detected at a single location, as a particle.
2) Quantum states are discrete. The "quantum" in quantum physics refers to the fact that everything in quantum physics comes in discrete amounts. A beam of light can only contain integer numbers of photons-- 1, 2, 3, 137, but never 1.5 or 22.7. An electron in an atom can only have certain discrete energy values-- -13.6 electron volts, or -3.4 electron volts in hydrogen, but never -7.5 electron volts. No matter what you do, you will only ever detect a quantum system in one of these special allowed states.
3) Probability is all we ever know. When physicists use quantum mechanics to predict the results of an experiment, the only thing they can predict is the probability of detecting each of the possible outcomes. Given an experiment in which an electron will end up in one of two places, we can say that there is a 17% probability of finding it at point A and an 83% probability of finding it at point B, but we can never say for sure that a single given electron will definitely end up at A or definitely end up at B. No matter how careful we are to prepare each electron in exactly the same way, we can never say for definitiviely what the outcome of the experiment will be. Each new electron is a completely new experiment, and the final outcome is random.
4) Measurement determines reality. Until the moment that the exact state of a quantum particle is measured, that state is indeterminate, and in fact can be thought of as spread out over all the possible outcomes. After a measurement is made, the state of the particle is absolutely determined, and all subsequent measurements on that particle will return produce exactly the same outcome.
This seems impossible to believe-- it's the problem that inspired Erwin Schrödinger's (in)famous thought experiment regarding a cat that is both alive and dead-- but it is worth reiterating that this is absolutely confirmed by experiment. The double-slit experiment mentioned above can be thought of as confirmation of this indeterminacy-- until it is finally measured at a single position on the far side of the slits, an electron exists in a superposition of both possible paths. The interference pattern observed when many electrons are recorded one after another is a direct consequence of the superposition of multiple states.
The Quantum Zeno Effect is another example of the effects of quantum measurement: making repeated measurements of a quantum system can prevent it from changing its state. Between measurements, the system exists in a superposition of two possible states, with the probability of one increasing and the other decreasing. Each measurements puts the system back into a single definite state, and the evolution has to start over.
The effects of measurement can be interpreted in a number of different ways-- as the physical "collapse" of a wavefunction, as the splitting of the universe into many parallel worlds, etc.-- but the end result is the same in all of them. A quantum particle can and will occupy multiple states right up until the instant that it is measured; after the measurement it is in one and only one state. 
5) Quantum correlations are non-local. One of the strangest and most important consequences of quantum mechanics is the idea of "entanglement." When two quantum particles interact in the right way, their states will depend on one another, no matter how far apart they are. You can hold one particle in Princeton and send the other to Paris, and measure them simultaneously, and the outcome of the measurement in Princeton will absolutely and unequivocally determine the outcome of the measurement in Paris, and vice versa.
The correlation between these states cannot possibly be described by any local theory, in which the particles have definite states. These states are indeterminate until the instant that one is measured, at which time the states of both are absolutely determined, no matter how far apart they are. This has been experimentally confirmed dozens of times over the last thirty years or so, with light and even atoms, and every new experiment has absolutely agreed with the quantum prediction.
It must be noted that this does not provide a means of sending signals faster than light-- a measurement in Paris will determine the state of a particle in Princeton, but the outcome of each measurement is completely random. There is no way to manipulate the Parisian particle to produce a specifc result in Princeton. The correlation between measurements will only be apparent after the fact, when the two sets of results are compared, and that process has to take place at speeds slower than that of light.
6) Everything not forbidden is mandatory. A quantum particle moving from point A to point B will take absolutely every possible path from A to B, at the same time. This includes paths that involve highly improbable events like electron-positron pairs appearing out of nowhere, and disappearing again. The full theory of quantum electro-dynamics (QED) involves contributions from every possible process, even the ridiculously unlikely ones.
It's worth emphasizing that this is not some speculative mumbo-jumbo with no real applicability. A QED prediction of the interaction between an electron and a magnetic field correctly describes the interaction to 14 decimal places. As weird as the idea seems, it is one of the best-tested theories in the history of science.
7) Quantum physics is not magic. [...] As strange as quantum physics is [...] it does not suspend all the rules of common sense. The bedrock principles of physics are still intact: energy is still conserved, entropy still increases, nothing can move faster than the speed of light. You cannot exploit quantum effects to build a perpetual motion machine, or to create telepathy or clairvoyance.
Quantum mechanics has lots of features that defy our classical intuition-- indeterminate states, probabilitistic measurements, non-local effects-- but it is still subject to the most important rule at all: If something sounds too good to be true, it probably is. Anybody trying to peddle a perpetual motion machine or a mystic cure using quantum buzzwords is deluded at best, or a scam artist at worst.

A: In plain English: Quantum physics is the realization that probabilities are Pythagorean quantities that evolve linearly.
That's it. All the rest is detail.
See slides 5-7 in Scott Aaronson's presentation: here.
