To understand black holes, we need to discuss the effects of gravity. In Newtonian physics, any two objects with mass will have an attractive force between them. This force goes as:
F = mM / d²
where m and M are the two masses, and d is the distance between the center of mass of the objects. As the product of the masses gets larger (or the distance between them gets smaller), the force gets stronger.

If we throw a ball into the air, we know it will fall back to us due to gravity. The higher the velocity, the higher it will go. Eventually, if you could throw it fast enough to reach "escape velocity," (~25,000 mph on earth) the ball could escape the earth's gravity field and would keep on going. As the gravitational force increases, the escape velocity also increases. Now, if we imagine an object with far more mass than the earth packed into a very small area, the escape velocity could be greater than the speed of light.

Black holes exist when a region of space is so dense that no object, not even light, can escape its gravitational pull. While the Newtonian view gives us a reasonable way to picture this, to really understand what's happening, we need to talk about Einstein's theory of relativity. According to relativity theory, gravity is the result of a curvature of space time . Massive objects in space distort (or bend) both space and time around it. For simplicity, let's not worry about time, but just think about bending space. Imagine the surface of a bed mattress as being a two-dimensional representation of space. If we place a bowling ball onto that mattress, the bed would be depressed (or bent) all around the bowling ball. The depressed part of the mattress is similar to what happens when space is distorted. That bending of space is what makes objects "fall"towards an object (in this case, the bowling ball) due to gravity. Let's now imagine rolling a marble just to the side of where the bowling ball is. Because the mattress is depressed around it, the path of the marble will be affected. If it is rolling fast enough, it will not fall into the ball, but its path will be "pulled" towards the ball by the curvature. This is what happens in space as light passes by massive celestial objects - for example, light can be bent around stars! The more massive (and dense) an object, the more space will be distorted around it. Black holes are so dense that space( and time ) is distorted to the point that light passing within a certain distance (the event horizon) will not be able to escape.

On a side note, black holes may not necessarily be so "black." Years ago, Stephen Hawking theorized that black holes could slowly leak radiation back into space due to virtual particles and vacuum fluctuations. If you're interested in following up on this, I'd suggest reading "A Brief History of Time" by Hawking.

The short answer is gravity. Think about the earth; because of gravity, we are held to the earth, and we have to move very quickly to escape the earth's gravity. A black hole is something with so much gravity that not even light can move fast enough to escape. And according to the special theory of relativity, nothing can move faster than light. The long answer is maybe too complicated for my response, but I hope you look it up and try to understand from a book!

Basically, general relativity says gravity describes the curvature of space and time; we feel attracted to the earth because that is the way space and time behave near the earth (this even affects the timing of GPS satellites, so we see relativity every day in GPS systems!). Near a black hole, space and time are very curved, and "inside" a black hole, they are so curved that nothing can get from inside to outside!