This is the briefiest of overviews of the General Theory of Relativity, and then necessarily also an oversimplification. Only the very basic ideas are laid out, and as simply as possible. The book contains a section summarizing the basic notions; as for the exam, there is a page with comments on the relevance of the book material on relativity theory. You can also look at an overview of special relativity, or at a very short summary of both.
Notions of space and time, regarded as something obvious and absolute in classical (Newtonian) physics, receive a very different interpretation in Special Relativity, where they are seen dependent on each other, the components of the entity of space-time. Separate notions of space and time have no definitive physical meaning, as they depend on the speed relative between observers, and it is only at our (low) speeds that our perceptions of them yield reasonable results. Further, mass and energy are found to be equivalent to each other, and energy and momentum are intertwined just like space and time are. All this holds for systems that move without acceleration (in "inertive frames of reference"). These statements of special relativity had marked a revolution in physics when they appeared, radically changing some of the foundations of physical science of the previous centuries.
Still, even in this theory, the space-time is taken as something given, existing as it is without regard to matter; and even though the old notion of absolute space and time is removed, some preference is still there: the theory deals only with unaccelerated systems (ones that move evenly, with a constant velocity). The energy, with its profound relation to mass, has no influence on the space-time. In the general theory of relativity all this is changed, and we go all the way; the restrictions are removed (the frames of reference that accelerate are OK), and the spacetime itself depends on the energy around. The behavior of such space-time (now it is not a fixed and unchanging thing) is seen as very complicated – or, in other words, our understanding of it gets much deeper.
The very basic ideas of general relativity can be stated so as to reflect two aspects of the theory. (These two follow from the so-called weak and strong equivalence principles. Note that they are not really independent, separate statements.)
Accelaration and gravity are equivalent. When in free fall in a field of gravity, we cannot observe the gravity (since we are accelerating accordingly). This is seen, for example, in the well-known fact that bodies with different mass fall evenly. This is not just a lone curiosity of our gravitation, but rather a direct consequence of the fact that the gravitational and inertive aspect of 'mass' happen to be equal; this observation can be taken to be the experimental motivation for the theory of general relativity. (Much like the constancy and uniformness of the speed of light was for special relativity.)
This is a fundamental statement about the nature of gravity and motion in it: somebody who is accelarating can simply consider themselves to be in a field of gravity (equal but opposite to the acceleration), and then they may forget about their motion and observe correct physical laws (with this gravity and without acceleration). Note that this statement, simple as it may appear, has profound meanings. (Remember how in special relativity observers moving relative to each other will see different distances and times ? Now, for example, an accelerated observer may see even different numbers of particles from what the unaccelerated one sees!)
The other 'aspect' goes in fact much further, and may be taken to be a definitive statement of our understanding of the space, time and 'gravitation.' To aid the arguments below, let us take an example of the electric field: its source is a charged particle, the field is different at different points in space (depending on where and how far from the particle one is), and is changing according to the state and motion of the particle that carries it. Our very spacetime is a physical quantity much like the electric field: its source may be taken to be mass, and the spacetime changes depending on how far from the mass one is, and rotates, moves and alike with the mass. Also, the spacetime depends on the mass in a very complicated way, and is really distorted ('curved') in the presence of the mass. Imagine a sheet of paper, that you take and crumble it in your hand; its surface is then not flat anymore. In a similar way, our space – and our time – are 'crumbled,' in a pattern depending on the mass around and the way it moves. (And of course, space and time are not separate entities but the components of the space-time, and thus the 'distortion' of each depends on the other, too.) This being the space we live in, it is difficult to imagine the effect, and what this – our – 'curved' space really looks like. (Under our conditions, of course, this effect is very, very small; otherwise we would be able to demonstrate it.) One way to ponder about this is by thinking about light beams. Light tends to go via shortest, straightest path; around a big mass, this is not going to be our usual straight line, and if we look from afar we will see that the light ray bends around. (And we do, in astronomy.) Now the kink with this is ... for that light beam, there is no 'distortion:' that is the way its space is, and thus the beam only follows what is the shortest path in its space – it is only for us, away from the influence of this mass, that its path is curved. Pretty much like the distances are different for observers moving relative to each other; but now no relative motion is called for: the space and time are different at different points in the field of this big mass. Put simply, you have an object in your hands, and you burn a match; you move around in the gravitational field, and this object's size is different, and the match burns at a different rate! (Talk about physical reality!) Finally, since mass and energy are equivalent, as established in special relativity, all the above statements about the mass affecting the spacetime hold for any energy; the light passing by also curves the space (but only very little). For one thing, this gives us a very different way of viewing the gravity and why things move the way they do: the space around objects – mass or energy – is curved (along with time), and thus every thing that moves around in this space really simply moves along its 'shortest paths.' (And when there is no mass or energy around, these 'shortest paths' are our usual straight lines.) So there is no need to talk about forces at all.
The general relativity is not only a "theory of gravity," but rather of spacetime, and of how it is affected by mass and/or energy. In a complete departure from the deepest presumtions of classical physics and our intuition, it states that the space-time does not simply exist regardless of anything, but is rather very readily affected by mass or energy (which can be considered its source), and behaves like any other physical quantity, changing around, following the behaviour of its source. (When we take into account the interpretation of mass in modern theories of physics, as due to interactions of 'fields,' we arrive at a view of physical reality which drops almost any reference to 'material particles;' in a curious accord with this, the equations of general relativity in fact do not operate with 'mass,' but with 'energy-momentum.' This point is summarized in section 11.5 (p300) of the book.)
To add a hint of concreteness to this story, here is the equation describing the
behavior of the spacetime, that has a similar meaning to the equations of mechanics
or electromagnetism: how does spacetime depend on its sources. (In fact, its
mathematical structure is just like the one of the equations of electromagnetism.)
It is really a compact presentation of a set of equations, called Einstein's field
equations:
Gμν - Λgμν = Tμν
The first quantity from the left holds the properties of the space time, the second is related to the so-called cosmological constant – while on the right-hand side (of the equality sign) we have the energy-momentum. Thus, even without any knowledge of mathematics, the first and simplest look at this tells us that the properties of spacetime are related to energy (and so on mass too); this is one of the most profound statements in science.
It should be stressed out that the effects of the described nature of space-time depend on the energy/mass content in such a way that they are extemely small for the weak gravitational fields, or small masses, that we have on earth. In this sense, the Newtonian description of our reality is acceptable, as an approximation, and the general theory of relativity does include this: when the smallness of masses is taken into account, the equations can be shown to reduce (to a very good approximation) to the usual Newton's laws. Of course, this is critical for the validity of the theory, since we know that Newton's physics does describe our world, at least to some accuracy. Note that this does not deny general relativity in our conditions; it only states that such effects are so small that we do not see them.