A black hole is a region of space that has so much mass concentrated in a small area that there is no way for a nearby object to escape its gravitational pull.
In order to escape from the Earth's gravitational pull, an object must accelerate away from the surface at least 11.2 km/s. This is called the escape velocity. Any slower and the object will fall back to the surface. The same thing is true for a black hole, however the escape velocity is so great that not even light has sufficient speed to escape.
Black holes are so called because we cannot see them; no light can leave its surface (either emitted or reflected) therefore it remains invisible. We can only detect black holes when they pass in front of another object and we witness gravitational lensing. To date, no black holes have actually been detected and confirmed, although there are several possible candidates.
It should be noted that contrary to popular belief, black holes are not "vacuum cleaners" sucking in everything around them. A black hole with a mass equal to that of the Sun will have a radius about 3 km. At a distance of about 106km, this black hole has no more attraction than any other body of the same mass. For example, if the sun were replaced by a black hole of the same mass, the orbits of the planets would remain unchanged. Once you cross the event horizon however, it is a different story.
When a star reaches the end of its life, it cools down and contracts under gravity. As it contracts it becomes more and more dense as the volume decreases. Some stars will explode (super novae) while others become neutron stars or black holes. Imagine a star 10 times bigger than our Sun, collapsed in on itself to the size of about 30km. The resulting object is so dense, its gravitational pull is immense.
Black holes could also be formed through high energy collisions, such as colliding neutron stars in binary systems.
Stephen Hawking provided a theoretical argument for the evaporation of black holes in 1974. The Hawking radiation process reduces the mass of the black hole and is therefore also known as black hole evaporation. Because Hawking radiation allows black holes to lose mass, black holes that lose more matter than they gain through other means are expected to dissipate, shrink, and ultimately vanish. Smaller micro black holes are predicted to be larger net emitters of radiation than larger black holes, and to shrink and dissipate faster.
Finding a Black hole
There are a number of theoretical techniques that can be used to locate black holes. Since a black hole cannot be directly observed, we must use indirect methods - looking for the effect of a black hole on its surroundings.
Accretion disks and gas jets
Massive, ultra-dense objects such as neutron stars and white dwarfs can cause accretion disks and gas jets to form, and it is believed that a black hole will behave in a similar way. We can see accretion disks and we can account for the central star, but they could identify where it might be worth looking for a black hole.
Extremely large accretion disks and gas jets may be good evidence for the presence of super massive black holes, because as far as we know any mass large enough to power these phenomena must be a black hole.
Strong radiation emissions
Intense, one-time gamma ray bursts may signal the birth of a black hole because astronomers believe that GRBs are caused either by the gravitational collapse of giant stars or by collisions between neutron stars. Both types of event involve sufficient mass and pressure to produce black holes.
In the image below you can see observed gravitational lensing as imaged by Hubble Space Telescope in the galaxy cluster Abell 1689.