Special relativity
Just as QM is the theory of the very small, Relativity is the theory of very large dimensions.
As usual, if this page gets to be too much, try the simpler version. This one is recommended, though.
In fact, there are two theories of relativity. The simpler, first one, called Special Relativity (SR), is about spacetime, relative movement and the speed of light. It says:
 the equations of physics are the same (invariant, in mathphysics speak) for all observers who are moving with uniform steady motion relative to each other^{1}Such observers are said to be in inertial systems.;
 the speed of light in a vacuum is a constant, the same for all observers.
The first statement, the principle of relativity, means that if someone goes by you in a highspeed train on a straight track, then either you or she can consider herself to be stationary and the other moving. If you have traveled by train, you may have wondered sometimes whether your train was advancing or the one next to you was moving backwards.
The second statement means that if both you and your friend in the train measure the speed of a light beam, you both will find the same answer, about 300,000 km/sec. This is not intuitive.
From these two statements, lots of things follow, including.

the equivalence of mass and energy, expressed by everybody’s favorite (and perhaps only) equation, E=mc^{2};

the fact that only bodies without mass can travel at the speed of light;

curious distortions like the faster someone is moving relative to us,

the heavier she gets,

the shorter she gets in the direction of motion and

the slower her clock runs, including her body clock, the heart.

This last point is the origin of the famous twin “paradox”. If your twin takes off in a space ship and travels fairly fast compared to the speed of light, then when she gets back to Earth, she will be younger than you, the twin she left behind. Strange, but this has been tested (with particles) and found to be true.
All this comes out of the mathematics for describing relative motion, which is in turn due to the fact that space and time are not independent, but form a fourdimensional “thing” called spacetime. What we see as space and time are aspects of spacetime and can appear differently for different observers. And they can get mixed up together, especially if you are traveling at high speeds.
General relativity
The other Relativity theory is called General Relativity (GR). Whereas Special Relativity is the theory of spacetime and light, General Relativity is the theory of gravity. GR says that spacetime is curved and that it is more curved where gravitational forces are stronger. In fact, gravity is the curvature of spacetime. Think of a plane surface with a depression in it. Put a ball on it and the ball will roll into the depression. Try to visualize that in four dimensions (Good luck!) and you’ve got GR. Uh, sortof …
There are also lots of things which follow from GR. One of the more interesting is that gravity can change the direction of light, since light travels through the space deformed by gravity. This and many other predictions have been observed to occur just as GR predicts.
Curiously enough, clocks run slower in a stronger gravitational field. A clock runs faster on top of a high building or in a satellite than it does on the surface of the Earth. For a satellite, this is opposite to the effect of high speeds on fastmoving clocks in SR, which makes them run slower. The two corrections are in opposite directions but do not cancel each other completely, so both must be applied to GPS systems in order for them to function correctly.
More recently, we have discovered that not only is spacetime curved, but that space is expanding. This is because a gravitational field exerts not just a force, but a pressure. This pressure is considered to come from a term in the equations of GR called the Cosmological Constant. Unlike the force of gravity, which is always attractive, the pressure can be negative, in which case it is not attractive, but repulsive. It is the outward force of the negative pressure which drives the expansion of space. For the last 7 Gy^{2}Gy means gigayears, or 10^{9} years or 10 billion years., the expansion of space has been accelerating. More about that in the cosmology chapter.
QM and GR — an unhappy marriage
As far as they go, SR and GR are as true to nature as can be detected within their separate domains, and so constitute bodies of knowledge – theories. They must be taken into account practically for the technology of GPS systems.
The problem is that QM does not work on the scale of GR, nor the other way around. This is probably the biggest problem in contemporary physics, the resolution of which is the grail now pursued by theorists.
In summary:
We live in a world where matter sometimes behaves like a particle and sometimes like a wave and where only probabilities can be calculated. All this is taking place in a curved fourdimensional spacetime which is expanding at an accelerating rate! And this space is not an empty vacuum, it is something.
Got that?
Now onto the the standard model of elementary particles.
Notes
1.  ↑  Such observers are said to be in inertial systems. 
2.  ↑  Gy means gigayears, or 10^{9} years or 10 billion years. 