A black hole is what remains when a massive star dies.
If you have read How Stars Work, then you know that a star is a huge, amazing fusion reactor. Because stars are so massive and made out of gas, there is an intense gravitational field that is always trying to collapse the star. The fusion reactions happening in the core are like a giant fusion bomb that is trying to explode the star. The balance between the gravitational forces and the explosive forces is what defines the size of the star.
As the star dies, the nuclear fusion reactions stop because the fuel for these reactions gets burned up. At the same time, the star's gravity pulls material inward and compresses the core. As the core compresses, it heats up and eventually creates a supernova explosion in which the material and radiation blasts out into space. What remains is the highly compressed, and extremely massive,core. The core's gravity is so strong that even light cannot escape.
This object is now a black hole and literally disappears from view. Because the core's gravity is so strong, the core sinks through the fabric of space-time, creating a hole in space-time -- this is why the object is called a black hole.
The core becomes the central part of the black hole called the singularity. The opening of the hole is called the event horizon.
You can think of the event horizon as the mouth of the black hole. Once something passes the event horizon, it is gone for good. Once inside the event horizon, all "events" (points in space-time) stop, and nothing (even light) can escape. The radius of the event horizon is called the Schwarzschild radius, named after astronomer Karl Schwarzschild, whose work led to the theory of black holes.