This is a long answer, but I hope you enjoy reading it. I'm going to try to explain completely for you, but not getting overly technical. Enjoy!
A black hole is an object that is extremely compressed. They're usually the remnants of a dead giant star. A living star is in a state of equilibrium - the force of gravity is crushing the star, trying to compress it into a smaller size, while the outward forces caused by the star's fusion reactions are trying to expand the size of the star. The star becomes the size where these two forces balance each other, and it becomes stable.
Once the star burns all of its fuel, there is no more outward force of expansion to balance gravity. Gravity then compresses the matter until one of a few things happens. If there isn't enough matter there for the gravity to be very strong, then eventually, the repulsive forces between the nuclei of the atoms in the material will balance gravity. At this point, the star is as compact as it can get, and will simply glow white hot like a coal for billions of years. This is a white dwarf star - our Sun is fated to become one of these.
If there is so much mass present that gravity is stronger than the repulsive forces between the atoms' nuclei, then it keeps collapsing down into a single point of infinite density. (called a singularity) This is a black hole.
Solid bodies can be thought of as having a "center of gravity." This is the one point of the body that all of its gravity seems to come from, and on which other bodies' gravity seems to pull the most. It's actually something that doesn't exist, except in engineering equations, etc - because the gravity of something comes equally from all of the matter at once, and not from a single point. Take the Earth, for instance. You're about 4000m above the Earth's center of gravity - that's how thick the Earth is from the center to the surface.
The law of gravity says that the closer you are to something, the stronger the gravity is. You would think, then, that as you tunnel into the Earth, since you're getting closer to its center of gravity, that the gravity would be stronger. It's actually weaker in there, because as you dig down, some of the Earth's mass is now above you, and the gravity from that part is cancelling out the gravity of the rest of the Earth below you.
So - as you can see, the strongest gravity is at the object's surface. This is true of stars as well. When the star collapses small enough, you can be "outside of it" and yet very close to its center of gravity. Being this close means that it's a lot stronger. If you crunched the Earth down to the size of a basketball, then it would still have the same gravity when you were 4000 miles away from the basketball, and thousands of times stronger gravity right at its surface.
There is a limit to how far this can go - it's called the Swarzchild Radius - the distance from an object where its gravity is so strong that even light does not go fast enough to get out. The point where this happens is called the event horizon - a shell around the black hole. It can also be thought of as the "surface" of the black hole, since that's where the "point of no return" begins. Outside of this, the object's gravity is normal, and can be escaped.
Inside of the event horizon, the star continues to compress itself down to that singularity that I mentioned earlier. What happens inside the black hole is still a subject of much debate.
Getting near one is very strange. The intense gravity warps space tremendously. You could see stars that are behind you when you looked at the edge of the black hole - the light from the stars that bareley misses the event horizon will circle around it, and escape again, coming back in your direction. Time slows down as you near the black hole as well, again due to the warping of space.
If you were to go into it, you'd be in deep trouble - the intense gravity would rip your body apart right at the edge. For the last few millionths of an inch, each millionth would be several times stronger gravity than the last. Your body, being a lot bigger than millionths of an inch, is not strong enough to hold itself together against this difference in gravity!
Imagine unravelling a sweater - that's about what would happen to your body as it crossed the event horizon.
So in short - a black hole is a star that got crunched into a very small space by its own gravity. It's made of the same stuff the star was when it burned out - a mix of elements like helium, oxygen, silicon, and iron.