A black hole is a region of highly curved spacetime. It has an event horizon which marks the point beyond which nothing can escape.
Specifically, space-time is so strongly curved that it is impossible for anything inside the event horizon to do anything but fall inwards.
Because black holes are so small for their mass, dust and debris attracted to the black hole tends first to gather in an accretion disk on the equatorial plane of the black hole before falling in.
Because the orbits are so small on the inner edge of the disk, the dust accelerates to a very high speed and acquires enough energy to give off light. Material can also be ejected at high energies (called x-ray jets) in a path pointing along the poles of the black hole.
A black hole can be formed when a star collapses, though it depends on the mass and composition of the star. When the nuclear fuel powering a star runs out, it can no longer sustain itself against gravity and it collapses. For less massive stars, the collapse may eventually be stopped at a point when the electrons or neutrons in the atoms of the star strongly resist being packed together, but if the star is massive enough it keeps collapsing.
For a black hole with the mass of the sun to form, it actually has to be packed into a sphere with a diameter of 6 kilometers!
Disclaimer: I'm doing most of this from memory, and it's been a while since I really studied this stuff, so I may have some minor details wrong!