How can time be relative? I don't understand how time can be relative to different observers, and I think my confusion is around how I understand what time is.
I have always been told (and thought) that time is basically a measurement we use to keep track of long it has been since an objects inception.
If that is even somewhat true, how can time be relative? If I have a rate of decay of X and you are somehow able to observe that (such as watching me age) how could I age at a different rate to 2 observers?
If time slows down the faster you go, does that mean you age slower? Or do you age at the same rate, only it seems like it takes longer? If a second is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom at 0K, how can that change at higher speeds? Assuming the temperature stays the same, shouldn't the measurement be the same?
Moving faster surely can't cause cells to decay slower, or atoms to radiate slower...can it?
Can someone explain, in the simplest of terms, how time can be relative?
 A: Intuition and perception (or the lack of there of) can be a big problem when you're trying to comprehend the implications of special/general relativity. You must understand that in everyday life which fuels our intuition is pretty slow. Most people don't move faster than $900 km/h$ or $250 m/s$. And that's a luxury for most, to travel by a fast jet. 
The speed of light is a staggering $299 792 458 m/s$. That is a million times faster from anything we have today. Just because time seems to be relatively absolute (pun intended) from our standpoint because our stage is rather small, the time it takes light to propagate from one point to another is so small, it doesn't mean that time is indeed invariant.
The interesting bit is that while Michelson & Morley were working on their amazing interferometer to measure Earth's speed in relation to the "magical aether", a man by the name of Hendrik Antoon Lorentz made a fascinating discovery about the nature of things, especially the nature of electrons. The direction-of-motion parts of the interferometer contracted as they moved and thus prevented any relative motion or interference to be detected. The two light signals always came at the same time because of the length contraction in the direction of movement.
Michelson couldn't accept this. It went against his life's work. Lorentz did the mathematical foundation as an explanation to the problem, but he did little to analyze the result. Einstein came to the same equations by following a different train of thought, this time involving the nature of Galilei-Newton relativity (which troubled him), the problem of light and all the evidence that was pointing that propagation through spacetime is constricted by a velocity limit. $299792458 m/s$. 
So, Einstein "took the bone" nature was throwing him. The speed of light is constant for ALL observers. No matter if they're sitting, falling, running, flying, sleeping. A light signal is exactly $c$ at all times. No matter how fast you're moving relative to others.
If that is true, then something else must bend. Space and time become intertwined to accomodate the nature of our existence, to allow light to travel at $c$ for all observers. From these simple postulates, which include the inability to differentiate intertial frames of reference, comes the death of simultaneity and of absolute time. 
A simple proof involves a moving light clock which passes exactly time $t$ in his up-down trip. When it begins to move with someone, a second observer - you on the ground will notice that its path elongates relative to you. Therefore, the time it takes to go up and down increases to $t_1$. All the while, the man on the moving platform sees the light perfectly in sync, up and down, because he's moving with it. Therefore, for him, you're the one who is slowed down in time (the $t_1$).
That's special relativity, but the one who really experiences slower decay or the relativistic effects is the man who is accelerating. So, yeah, time is relative to protect the constancy of the speed of light.
Hope it helps. And give it time. It has been proven many times over and a lot of scientific work today relies on relativistic effects of time dilation.
ADDENDUM:
It's exactly the repercussion of this. Decay is the passage of time. The biological processes are the same, but if he is moving really fast (and let's say uniformally), to every other observer the time slows down for the man onboard (lightclock thought experiment, proved with satellite sync and plane/atomic clock experiments). Also, to every other observer the ship contracts. To the man onboard, he feels nothing. The passage of time is the same and the ship dimensions are the same. To protect relativity, he sees that others are slowed down in time and contracted. But he is the one who accelerated, therefore, he is experiencing the time dilation.
And thus, time dilation implies slower time passage for the man onboard relative to other stationary observers. He feels normal, relative to him, time runs "nicely", but when he comes back, the relativistic effects will have done their part based on the famous $\gamma^{-1} = \sqrt{1−(v/c)^2}$. This has not yet been proven directly, but it is inferred from various experiments done by planes and atomic clocks and also the need to sync up satellites after a while because of the gravitational differential. Why? Time goes slower, decay is dependent on time, slower decay.
The final and most important point would be that time passes for everyone in the same manner (you can't feel a change). But it is the relativity (comparing to someone else) which enables us to detect time passage differences. Just like you can't know how it feels to be a rabbit, because you've never had a chance to be one to make the comparison. A blunt, but accurate comparison. Just like you can't imagine a different kind of existence because you can't compare to an another Universe (we've never been in it). That's the gist of relativity. Everything we know is relative. That's "how" we know. 
But the beauty of the human mind and the triumph of all science lies in the fact that we can contemplate this, our own limitations, our ways of thinking. And by doing so, we find a way to overcome them or to make the most of them.
A: You're "understanding of time" is perfectly okay. What you are missing is the relation between space and time. If you (an observer) are sitting (flying) there, looking just at your clocks -- then you will never notice any time dilation. 
Relativistic effects appear when we have different observers in different points of space. Moving around, observing, sending signals, e.t.c. And when this observers try to make a coherent picture of their observations, they arrive at the conclusion that one cannot extend his local "understanding of time" to the "understanding of time" for the whole space.
Actually space and time turn out to be very strongly interwoven, so we usually prefer to use the term spacetime.
A: So, here's the deal. "Time is relative" means a lot of different things to a lot of different people. In order to make a solid step forward, Einstein and company basically needed to clarify what they were trying to say.
What they were trying to say looks something like this: "if you see a train passing by you, you're going to see things happen in slightly "fast motion" when it's at a distance coming toward you, and in slightly "slow motion" when it's at a distance moving away from you. This is no surprise; when you hear the train blow its horn it sounds higher-pitched as it approaches and lower-pitched as it leaves. BUT, when you successfully correct for this 'Doppler effect,' as the speed of the train gets closer to the speed of light, you will find that actually, in your coordinates, the train and things happening inside of it appear to happen in uniformly "slow motion", slowed down by the factor $1/\sqrt{1 - v^2/c^2}$."
As you can imagine, this effect is a little hard to observe! It really comes from "adding up" a lot of little effects that happened when the train accelerated to this massive speed.
The most obvious little effect is that when you accelerate in the $x$-direction, changing your speed by a small amount $\delta v$, your sense of space coordinates changes to $x \rightarrow x' = x - t ~ \delta v$. That is, a wall that you used to think was a constant "5 feet away" (in the x-direction) is now going to start off being "5 feet away" but after a little time it might be "4 feet away", then "3 feet away", then "2 feet away", and so on. This is very obvious and was known to Galileo and Newton.
But there's a subtle effect, too, about time. Suppose you have clocks on two walls, one is $x = +5$ feet away, and one is $x = -5$ feet away. This effect says that they go out of sync a little bit, $t \rightarrow t' = t - x ~ \delta v / c^2$. The $c^2$ is a huge number that historically made this property of acceleration totally ignorable. But we can't ignore it so much these days, not with high-velocity particles that we have to calculate.
Now it turns out that you have to break apart what's happening into little intervals of time when you're accelerating, but if you add this little change up, many times over, then it says both that stuff happening in your train appears to be in slow motion relative to people outside the train, but also that stuff happening outside the train appears to be in slow motion relative to what's happening in the train. So it's because time has this little "we start to disagree on the simultaneity of remote events" property, that we eventually build up a larger "we start to disagree on how big things are and how fast their clocks are ticking" discrepancy. And the great part about it is: you're both correct. Both of you have perfectly valid coordinates which perfectly describe the world.
In fact, many contemporaries of Einstein thought that the newer "electrodynamics" science which was implying these things was fundamentally broken. Before Einstein, people knew about these problems due to a guy named Lorentz, but didn't tend to take his work too seriously. Einstein's 1905 paper said, effectively, "we have to take him seriously." 
One reason that we can now appreciate is: we now know that the mathematics is totally self-consistent. Nobody ever disagrees on the order of events and nobody can use these weird "de-synching clock" effects to travel in time, unless they somehow find a way to move faster than the speed of light. There is another reason why we think that nobody can move faster than the speed of light, which has to do with the key fact about these de-synching clocks: they coordinate together with the change in your spatial coordinates to ensure that once you stop accelerating, you still think that light moves every direction at constant speed $c$, even in your new $(x', t')$ coordinates. This means that if you challenge someone to race a beam of light, and they start moving at speed $c/2$ relative to you, the light is not moving at $c/2$ away from them in their coordinates, but at speed $c$ away from them. So there's a real Zeno Paradox here guaranteeing that nobody can ever outrun light.
The most obvious paradox which turns out to not be a big deal is, "if I think that the people on the train are moving in slow motion, and they see me in slow motion, can't I just call one of them up and we'll see who's faster and who's slower on the phone call?. And the answer to that is, yes, if a phone could transmit information instantaneously, then nature would have to establish one of these people as correct and one of them as incorrect. But, of course, real phones are also bound to transmit energy no faster than the speed of light -- and this gives the precise ambiguity that you need to make sure that both of them are perfectly correct and neither one can claim supremacy over the other.
So that's what we mean by "time is relative": someone on the street, after correcting for Doppler effects, still thinks that people on the train are moving "in slow motion" and thus aging slower than people on the ground. People in the train of course see themselves just fine, but after correcting for Doppler effects think that people on the ground are moving "in slow motion" and thus aging slower than the people on the train. Both groups have valid coordinates, and we cannot choose between them. Whenever you find an experiment which actually seems to test it, like "Well we'll stop the train and get out and check their ages," those coordinate shifts invariably balance everything out so that there is no paradox: usually the person who is accelerating becomes "wrong", so if we speed up to jump on the train we see the people on the train moving in fast-motion until they appear to be older than us, and it turns out that the "train was right"; but when the train slows down to check on the ages of the people on the ground, all those people move in fast-motion until they appear to be older than the people on the train, hence the "ground was right." Neither was really absolutely right, but there is a consistent math where remote clocks which seemed in-sync suddenly get a little out-of-sync, causing systematic disagreements about how fast clocks are ticking in general.
A: I'd like to add a few words to Domagoj Pandža's excellent answer. He makes this statement:

"....Intuition and perception (or the lack of there of) can be a big problem when you're trying to comprehend the implications of special/general relativity ...."

I think Domagoj's answer is excellent, but I disagree a little with this statement. Actually almost all of our intuitions about time survive special relativity: that's for me the most remarkable thing of all. What relativity teaches us is that there is more than one valid extrapolation from our everyday physical intuitions that is consistent with them: our intuition is sound, it's simply that the first guess at an extrapolation from it, namely Galilean relativity with its vector addition of relative velocities between inertial frames, is not correct. This is the unique relativity that can follow from Galileo's relativity principle- that no experiment from within an inertial frame can detect the frame's motion from observations within the frame alone - if we assume that all observers measure the same time interval between any two events in spacetime. But Occam's razor sometimes fails, and if we move onto the next simplest alternative and relax the assumption of absolute time, then there are actually other possible relativities that are all consistent with Galileo's and Copernicus's basic principles: these are the Lorentz transformations and the parameter $c$ simply determines which of this family of reltivities applies to our universe.
But relative time: surely that's against all of our intuitions? Let me try to convince you otherwise.
Although the relative time that Domagoj Pandža discusses shows that simultaneity is relative and that different observers will measure different times between two events, nonetheless the Lorentz transformation that defines these relativities is a very special transformation, such that most of time's important intuitive purely physical properties are altogether unchanged, utterly so. What are these basic intuitions? For me, they are (1) rhythm and rates of natural processes around us relative to one another and (2) causality: i.e. the effects of any causal agents we witness around us always seem to come after their causes in our universe.
Special Relativity changes none of these things even though time's numerical value is relative. That's how special the Lorentz transformation is.
So if the solar system happens to be moving through space at a rate relative some other celestial body at speeds approaching the speed of light, then this has no effect on the relative rates of progress of physical processes that happen around us. Our bodies experience circadian rhythms and things happen to our bodies - we grow from children, enter puberty, grown into adults and exorably grow old and die: at rates that bear certain, repeatable relationships to our Sun's apparent diurnal motion as do important natural events such as the seasons. So the time relationships between us and the physical processes around us are all the same. So time "feels" like it passes normally to us  when we refer to things in the World which are all still relative to us or moving very slowly. This is basically Galileo's principle. All of this "normality" prevails even though our being in the far off celestial body will observe our physical processes: ageing of biological beings, decay of metastable particles and so on to be happenning very slowly relative to their local physical processes owing to our slower relative time relative to theirs. And, if we build advanced telescopes, we'll actually observe their physical processes moving slowly relative to ours too!
This sounds weird, but here is the clincher. Because the Lorentz transformation is such that no cause-effect relationship can propagate at faster than $c$, we cannot instantaneously compare notes on what is happenning in our local frames. The delay in any signalling between the two frames prevents any contradition arising from the mutual time slowing of each frame relative to the other. This is because the Lorentz transformation gives us something even beyond Galileo's principle: for even though we shall disagree with our far off, relatively moving observer about even the order wherein things we observe each other's frames happen, we shall never disagree about the order of causally related events. Causes never come after effects in any inertial frame we observe, no matter how it may be moving relative to us. Our physical intuition of causality is beautifully respected by special and general relativity. The topology of the web of causal links between physical events in spacetime is utterly unchanged even though the web can be stretched and squeezed a bit when we look upon it from different inertial frames. The most basic intuition of all - causality - survives.
So in summary, our physical notions of time: how fast our bodies change, day and night and the seasons are not changed and any causal relationship we can observe are unchanged by relativity. We only see tiny differences when we begin to assign numerical values to the rates of these processes and we measure them with clocks that only the highest technology can give us. Nonetheless, all causal relationships hold true no matter how severe special relativistic effects become. 
I say more about all these things in my paper which I hope shall be published soon: I have submitted it to the European Journal of Physics. A preprint of it is here:
Rod Vance, "Of Groups, Galileo and What's So Special About the Speed of Light"
I also give a summary in my recent answer to the Physics SE question "If all motion is relative, how does light have a finite speed?"
A: The relativistic effects are real and have been confirmed experimentally. The universe actually obeys general relativity which exactly resembles special relativity in the absense of a gravitational field and very closely resembles it in Earth's gravitational field. According to special relativity, all objects have their internal clocks slow down by a factor of 1/sqrt(v^2 - c^2) and contract in length by a factor of 1/sqrt(1 - v^2/c^2) in the direction of motion. In fact, the theory of special relativity also predicts that there's no way to detect absolute motion. My answer here shows how that can be despite time dilation. According to general relativity, stationary clocks tick faster when they're higher and the rate of change of ln(time contraction) with height is g/c^2 where g is the gravitational field strength.
A: If you think about it, time as we know it does not actually exist/flow - it is our mental manifestations of the world around us that we think of as time. For example, what we see is not actually there as we view it. The object sends us light-waves (only a small portion possibily of what the object really is), our eyes then have to decode the light waves and our brain must decoded the neuronal signal from our eyes. Therefore, there are many levels of filtering that go on when we try to understand the universe.
That filtering makes it difficult to understand the passage of time. Time doesn't actually pass. Time doesn't actually exist and it definitely doesn't flow or pass. What time is - is just the change of particles relative to other particles. Our brain makes sense of the changing environment by the passage of time causing things to change, but really the rate at which things change relative to other things is making our brain conscious of time.
To that end, we must think about the particles - when there are a lot of particles close together in a confined space they slow down because they can't move anywhere and other forces prevent them from changing shape as fast as they would otherwise. Therefore, a mass heavy object is moving slower relative to other objects because the particles aren't changing as fast relative to particles external to the object. This is essentially what Einstein was saying: objects with great mass move slower than objects with less mass. Einstein also understood that at some threshold the energy expended in making particles change shape also can be converted to energy that causes particles to move (E=MC2). Therefore, a particle moving is expending more energy than another particle of equal mass at a standstill and thus is moving through time faster as well. Therefore, masses being equal between two objects the one moving faster will be moving faster through time to the other object.
