At last, we get to the fun part of astrophysics!
Anatomy of a Black Hole
A black hole is an object which has a gravitational pull so strong that not even light can escape it. As you approach the centre of a black hole, the curvature of spacetime - and thus the gravitational field strength - approaches infinity. In other words, a black hole has an infinitely deep potential well.
Black holes are formed when a star with sufficient mass dies - when it becomes unable to sustain nuclear fusion, its own gravity causes it to collapse in on itself and implode, causing a supernova. The remaining mass condenses into a very small space, leaving behind a black hole - or a neutron star if it doesn't quite have enough mass.
The singularity is the centre of a black hole, where all of its mass is condensed into an infinitely small point. A lot of our physics breaks down when trying to grasp the properties of a singularity (for example, quantum mechanics fails to describe what happens at a singularity and is an incompatible theory) - the very concept of an infinitely small mass has all sorts of complicated side effects, such as the idea of infinite density. Astrophysicists theorise that the singularity may be an infinitely thin "ring singularity" as opposed to a zero-dimensional point in space, as that allows us to explain the rotation of a black hole around an axis (due to conservation of angular momentum when a black hole is formed). The singularity is the heart of the black hole, and it's the only "real" part of one - everything around the singularity is just warped spacetime caused by the singularity.
The event horizon is the sphere around a singularity which light cannot escape from. As we know from the equivalence principle, the path taken by light can be curved by gravity, and within the event horizon this curvature is so significant that it is impossible for light to escape the black hole's potential well.
The Schwarzschild radius is the distance from the singularity of a black hole to the event horizon - it's the radius of the event horizon sphere.
is the Schwarzschild radius, in metres. is the gravitational constant, approximately . is the mass of the black hole, in kilograms. is the speed of light in a vacuum, in metres per second.