Let's start with Special Relativity, because it is the theory that Einstein discovered first. As we all have learned in our high school and college physics classes, Newton came up with a bunch of equations that explain how gravity affects things. He was able to demonstrate that gravity is a force that causes objects to accelerate towards one another at a constant rate (the objects don't move at a constant rate - their increase in speed over time is constant). He was also able to demonstrate that the acceleration of objects towards the earth was constant regardless of the weight of the objects. That is to say, a feather will drop to the ground (in a complete vacuum - air resistance obviously would slow it down) at the same rate as a bowling ball.
Why do objects fall at the same rate regardless of weight? The answer is fairly simple. Gravity is a force that is proportional to the masses of the two objects and inversely proportional to the distance between them. Thus, there is a stronger force of gravity between more massive objects than there is between less massive objects. The trick is that there is also more mass to pull, and so when you work it out to find out how quickly the objects accelerate towards each other, the masses end up cancelling out and you are just left with a constant. That constant is 6.674 x 10^-11 N*kg^-2*m^2. The units of the constant aren't important because they cancel out when you apply that constant to determine other things, but it should tell you that gravity is a fairly weak force. It is a force, however, that accumulates as more and more mass is applied, so eventually it can overwhelm the other forces.
Newton discovered a problem, however. His equations predicted the motion of lots of objects that were affected by gravity, like planets, but he discovered that the orbit of Mercury didn't entirely line up with his predictions. This wasn't enough of a problem to cause Newton's laws to be thrown away, but it did cause a lot of folks to wonder if it wasn't the complete picture. There were also some problems discovered later when scientists noted that the speed of light was measured to be the same regardless of how fast they were moving relative to the light source.
Einstein puzzled on these problems for a while before he came to the realization that Space and Time are connected. He discovered that motion in space and motion in time are not independent. The best way to think of this is to imagine you are driving down the highway and steer onto an offramp. As you do so, your motion in one direction will slow down, and your motion in the orthogonal direction will increase (if you were traveling north, your northward speed goes down and your eastward speed goes up).
Space and time work like this. When we are at rest, we are essentially moving through time at the speed of light (more on that in a minute). As we begin to move in space, our motion in time begins to go down. Time slows down. This has been proven by placing atomic clocks (which are extremely accurate) on planes and flying the planes at different speeds. When the clocks are brought back together, they show different times.
Another weird effect of this is that no matter how fast you are going, light will always appear to move away from you at the speed of light. If you got into a rocket and chased a beam of light, you wouldn't be able to catch it, no matter how fast you go. As you speed up, time in your reference frame will slow down while it continues to flow at normal speed for the beam of light. Whatever distance you gain on it is lost due to the slower time.
This, in a nutshell is special relativity. It isn't the whole picture though, because when Einstein came up with this theory, he didn't incorporate accelerated motion into it, just constant motion. In other words, he couldn't incorporate forces like gravity that cause you to accelerate. He had to go back to the drawing board and come up with a new thoery to incorporate accelerated motion. I'll talk about that next.