In order for a star to remain a star, it needs to be in this constant state of equilibrium, where there are some forces (mostly fusion reactions) that make it expand, and some other forces (gravity) that make it contract. When all forces are balanced, the star maintains its size. But the problem starts when the star runs out of fuel to continue to produce nuclear fusion reactions. What happens then is that the inward-pulling forces win over, and the star starts to collapse.
Now, as it becomes smaller and denser, new forces start playing important roles. These are more complicated ideas - things like electron degeneracy pressure - so I won't go into detail about those. The point is that, for some stars, these new forces become large enough that a new balance is found between the inward pulling and the outward pulling things.
In the case of black holes, even these new forces are not enough to balance things, so the star continues to collapse and become denser and denser. Mathematically, we say that it becomes as small as a single point. Of course, physically we would like some better explanation than that, but the problem is that we don't really know how the laws of physics are altered when you reach such intense density.
Now, you've probably heard that even light can't escape a black hole (which is why we call them black holes). The idea behind that is the following. If you want to orbit a star, you need to be moving pretty fast. In fact, the closer you get to the star, the faster you need to move if you don't want to fall in. I should add that this is the case with satellites orbiting Earth, too. As soon as they lose speed, they start spiraling onto Earth. Now, the speed at which you need to go in order not to fall it depends on how far you are from the star, and how much mass the star has. In the case of most stars and other objects, the speed of light is much much faster than the minimum required to not fall it, at any distance. Black holes, however, are so very very small (so you can get closer to them without being *at* them) and so very very massive, that the speed you'd need in order to not fall in (at a given distance) would be faster than the speed of light. But it's a law of nature that nothing goes faster than the speed of light! That means that, at a certain distance, everything would eventually fall into the black hole. Of course, you can always go further away from it, where the speed necessary to orbit is smaller, and you wouldn't fall in.