Black holes are objects so dense that not even light can escape their gravity.
Black holes are predictions of Einstein's theory of general relativity. The simplest static and spherically symmetric solution to Einstein's equations is the Schwarzschild metric, which describes the curvature of spacetime in the vicinity of a nonrotating spherical mass. This solution is only valid outside the gravitating object. However, if the radius of the object is smaller than a value known as the Schwarzschild radius, it becomes a black hole.
Below the Schwarzschild radius, spacetime is sufficiently curved that light rays always travel towards the center of the system, regardless of the direction in which they were originally emitted. Because relativity forbids local superluminal velocities, nothing can escape from within the Scwarzschild radius. In particular, the gravitating object will shrink into a point singularity.
The generalization of the Schwarzschild radius is known as the event horizon. More general black holes can be described by other solutions to Einstein's equations. An example is the Kerr metric for a rotating black hole, which possesses a ring singularity.
Black holes demonstrate some counter-intuitive properties of general relativistic spacetime. Consider a hapless astronaut falling radially towards the center of a Schwarzschild black hole. The closer she comes to the event horizon, the longer the photons she emits take to escape to infinity. Thus, a distant observer will see her falling slower and slower towards the event horizon, never reaching it. However, the astronaut, in her own reference frame, crosses the event horizon and reaches the singularity in a finite amount of time.
Black holes are also interesting when applied to other physical theories. A commonly stated proposition is that "black holes have no hair," meaning that they have no observable external characteristics that can be used to determine what they are like inside. Black holes have only three measurable characteristics: mass, angular momentum, and electric charge, and can be completely specified by these parameters.
The entropy of black holes is a fascinating subject. In 1971, Hawking showed that the total event horizon area of any collection of classical black holes can never decrease. This sounds remarkably similar to the Second Law of Thermodynamics, with area playing the role of entropy. Therefore, Bekenstein proposed that the entropy of a black hole really is proportionate to its horizon area.
In 1975, Hawking applied quantum field theory to a semi-classical curved spacetime, and discovered that black holes can emit thermal radiation, known as Hawking radiation. This allowed him to calculate the entropy, which indeed was proportionate to the area, validating Bekenstein's hypothesis.
In fact, it turns out that black holes are maximum-entropy objects: the maximum entropy of a region of space is the entropy of the largest black hole that can fit into it. This leads to the holographic principle.
Black hole entropy is an area of active research. See string theory.
There is now a great deal of observational evidence for the existence of two types of black holes: those with masses of a typical star (4-15 times the mass of our Sun), and those with masses of a typical galaxy. This evidence comes not from seeing the black holes directly, but by observing the behavior of stars and other material near them.
Galaxy-mass black holes with 10 to 100 billion solar masses were found in Active Galactic Nuclei (AGN), using radio and X-ray astronomy. It is now believed that such supermassive black holes exist in the center of most galaxies, including our own Milky Way.