This is a very brief and simplified overview of the Special Theory of Relativity; of course, a fairer treatment would assume an incomparably more content and subtlety. More or less at the same level, but with a lot more detail, similar material is covered in the book in the two chapters on relativity theory; as for the exam, there are pages with comments on the relevance of the book material on relativity theory. You can also look at an overview of general relativity, or at a very short summary of both.
Space and time are notions that we take for granted, as something obvious, simply being there. Until around the beginning of the 20th century, even though (rare) leading physicists had been expressing thoughts on their nature, classical physics too was based on space and time as something given, absolute and same for all, never changing. This is how our intuition works; the road may be bumpy and we may be bored, but a mile is a mile and an hour is an hour.
As it turns out, these very basic and deeply rooted premises are false; when looking at (very) high speeds and/or hard acceleration or very strong gravity, we can observe phenomena, by now many, that cannot be explained by physics based on this usual perception of space and time. The theories of special and general relativity deal with this, and offer treatments that radically change everything about concepts of space and time, and that have been confirmed by experiments too numerous to count. Some elementary statements of special relativity are given here. In order to carry out a few arguments let us use a following picture: we are cruising down a straight highway, passing a friend who is standing on the pavement.
In terms of the classical picture (based on Newtonian mechanics), this is a rather simple proposition. For us, she appears to be retreating, and if we label her position (relative to our car), we see that she is being further and further away, and it is a trivial matter to express this in numbers (using our speed and time elapsed). Also, as we are passing her we can pick a nearby sign and (somehow) measure how far she is from it; as we drive on we can measure this distance again, and will of course get the same number. More importantly, our friend on the pavement will see nothing wrong with our numbers: she measures the distance between her and the said sign just as we do, and gets it to be the same. (Let us say it is 10 meters.) As for times ... we can watch her burn a match, and see that it took 10 seconds; of course, she will have measured the same time. She stands 10 meters away from the sign, the match burns in 10 seconds, and that's it.
Now, at speeds we experience, everything appears to be fine with this picture: of course we measure the same distances, and of course that 10 seconds is 10 seconds, regardless of ‘who times it.’ But, if we were to be moving really, really fast (like rockets move, and then much faster) ... then we could easily find out that our perceptions of space and time, embodied by this picture, are plain wrong. So let us now assume that we can go that fast, and that we do; let us look at our picture again.
The distances that we measure, and the passage of time that we experience are different from what our friend standing on the pavement knows: for us, the distance from her to the sign is shorter than for her – if we see it as 10 meters she measures it larger; if we timed her match burn in 10 seconds – her clock showed to her less than 10 seconds. (In order to actually measure distances and time, we must send some signals that are then returned to us. Let us say we send two light signals, to her and to the sign, that bounce back to us, and we time their trips; then we can work out the distance. Please note: since we are moving away, clearly this will affect the difference in times for signals to her and to the sign, as the signal does take time to travel, and this is important ... but this is not a ‘reason’ or an ‘explanation’ for our different lengths and times: the distances and times for us and her, who are moving relative to each other, are different.) Just how much different the numbers are depends on the speed (relative between us) – the closer to the speed of light it is, the bigger the difference; for our typical speeds, this effect is so small that we cannot observe it.
In one sentence, for observers moving relative to each other, time runs differently and distances are different: the same object will have different lengths, and the same process will take different time, when measured by us and by our friend on the pavement. To add trouble, the mass of an object is different too (as measured by observers moving with respect to each other). Please note: these quantities do not appear to be different, and there is nothing inaccurate or subjective involved; the actual physical distances, time intervals and mass are different, for us and those moving relative to us.
All this is directly derived from two postulates, taken to characterize the special theory of relativity:
Along with these two postulates, which reflect the historical motivation for the theory, and with the unintuitive effects mentioned, there is a much deeper lesson here. Space and time are related, intrinsically connected to each other; time does not run independently of where something is happening, nor are distances (correctly) measured regardless of times at which they are measured. (As mentioned above, measuring the distance between our friend and the sign, while we are moving relative to her, involves receiving signals back from the sign and from her – and these will reach us at different times. So the time is involved!) This is the meaning of the fact that the relative motion between observers implies that distances, times and mass are different in their frames: space and time are not independent of each other; we must always take into account both, as coordinates on a completely equal footing. (In this sense, the speed of light has a role of ensuring that this 'equality' of space and time coordinates is balanced numerically in formulas, and takes care of units.) There is really one entity, "space-time." Its structure is such that our naive, everyday notions of distances and times do not have any definitive, independent meaning; there are somewhat more complicated quantities that do have such a meaning, and there are rules for dealing with those that don't. But we have to be careful with what we measure and use, and how; usual ‘space’ and ‘time’ lose their absolute, distinguished role. Again, this structure of space-time and its consequences, like effects mentioned above, cannot be observed at speeds that we encounter; they become noticeable at speeds comparable to the speed of light. (And this is fast: the light goes around the Earth in much less than one second!)
There is another deep statement of the special relativity: mass and energy are related, just like space and time are; they are in fact shown to be equivalent. (This stems from the fact that mass is measured differently in frames moving relative to each other; in addition to distances and time intervals, it is yet one more quantity, previously assumed absolute, that in fact does not have a direct physical meaning, since it is dependent on who measures it. But when energy is taken in concert with momentum, in a manner very similar to how space and time are ‘mixed,’ we get a very meaningful, fundamental quantity.) Note here that the relation between energy and mass has nothing to do with things moving and thus having kinetic energy, or similar; this equivalence involves mass and energy as fundamental quantities. One direct consequence is that matter can convert to energy, and energy to matter; there is an exact relationship, striking in its simplicity, relating the mass of this matter to the energy: E = mc2. This is a very far reaching principle, extending to other areas of physics in fundamental ways; using the concept of field, together with quantum mechanics it gives rise to quantum field theory, thus being an essential ingredient in modern theories of the structure of matter and interactions.