Black holes are objects of extreme density — they have a very high concentration of mass and an intense gravitational field. Black holes are not empty voids in space, but their gravity is so strong that nothing can move fast enough to escape their pull, not even light. Because light cannot escape a black hole, it is impossible to directly see one — that's why they are called "black." However, surrounding material that interacts with a black hole offers indirect evidence. For example, Doppler measurements of the fast orbital motions of gas and stars around a galactic center indicate something very massive located in the middle is accelerating the speed of the gas. The extremely high speeds could only be caused by something with an intense gravitational field, such as a super-massive black hole.
Black holes can exist in an infinite range of sizes, but in astronomy the focus is generally on two types. Stellar black holes — which are formed by the death of high-mass stars — have the size of a few solar masses. A solar mass is defined as the mass of the Sun, about 2 x 1030 kg (4.4 x 1030 lb), or roughly 330,000 times the mass of Earth. Super-massive black holes are the size of millions or billions of solar masses and are found in the centers of galaxies.
A super-massive black hole acts like a "monster" at the center of a galaxy: through the intensity of its gravitational pull it "swallows" material that will thereafter never be seen again. The animation in this resource is an artist's rendering of what it might look like to make the impossible trip to a super-massive black hole. From a distance, there is no evidence of a black hole — only the typical spiral arms and central bulge of the spiral galaxy are visible. As you sail into the bulge, you pass by individual stars on your way towards the center. When you reach the center, you see a swirling disk of gas and dust — the accretion disk of the black hole. Looking directly into the hole from above, you can see its central "blackness." However, although the black hole itself is the absence of light, the behavior of the surrounding material is a clear indication of its existence. The matter that surrounds the black hole rotates faster the closer it is to the center, and because friction from the high-speed motion of the gas and dust releases enormous amounts of energy as light, it also grows more luminous.
To learn more about the physics of black holes, check out String Theory: Strings to the Rescue.
To learn more about spiral galaxies, check out Spin a Spiral Galaxy.
To learn more about light, check out Speed of Light: How Fast Is That?.
To learn more about the Doppler Effect, check out Stellar Velocity: The Doppler Effect.