Minor update 2003 by Don Koks.
Original by Matt McIrvin.
In 1916, when general relativity was new, Karl Schwarzschild worked out a useful solution to the Einstein equation describing the evolution of spacetime geometry. This solution, a possible shape of spacetime, would describe the effects of gravity outside a spherically symmetric, uncharged, nonrotating object (and would serve approximately to describe even slowly rotating objects like Earth or the Sun). It worked in much the same way that you can treat Earth as a point mass for purposes of newtonian gravity if all you want to do is describe gravity outside Earth's surface.
What such a solution really looks like is a "metric," which is a kind of generalization of the Pythagorean formula that gives the length of a line segment in the plane. The metric is a formula that may be used to obtain the "length" of a curve in spacetime. In the case of a curve corresponding to the motion of an object as time passes (a "timelike worldline"), the "length" computed by the metric is actually the elapsed time experienced by an object with that motion. The actual formula depends on the coordinates chosen in which to express things, but it may be transformed into various coordinate systems without affecting anything physical, like the spacetime curvature. Schwarzschild expressed his metric in terms of coordinates which, at large distances from the object, resembled spherical coordinates with an extra coordinate t for time. Another coordinate, called r, functioned as a radial coordinate at large distances; out there it just gave the distance to the massive object.
Now, at small radii, the solution began to act strangely. There was a "singularity" at the center, r = 0, where the curvature of spacetime was infinite. Surrounding that was a region where the "radial" direction of decreasing r was actually a direction in time rather than in space. Anything in that region, including light, would be obliged to fall toward the singularity, to be crushed as tidal forces grew beyond limit. This region was isolated from the rest of the universe by a place where Schwarzschild's coordinates blew up, though nothing was wrong with the curvature of spacetime there. (This was called the Schwarzschild radius. Later, other coordinate systems were discovered in which the blow-up didn't happen; it was an artifact of the coordinates, a little like the problem of defining the longitude of the North Pole. The physically important thing about the Schwarzschild radius was not the coordinate problem, but the fact that within it the direction into the hole became a direction in time.)
Nobody really worried about this at the time, because there was no known object that was dense enough for that inner region to actually be outside it, so for all known cases, this odd part of the solution would not apply. Arthur Stanley Eddington considered the possibility of a dying star collapsing to such a density, but rejected it as aesthetically unpleasant and proposed that some new physics must intervene. In 1939, Oppenheimer and Snyder finally took seriously the possibility that stars a few times more massive than the Sun might be doomed to collapse to such a state at the end of their lives.
Once such a star gets smaller than the place where Schwarzschild's coordinates fail (called the Schwarzschild radius for an uncharged, nonrotating object, or the event horizon), there's no way it can avoid collapsing further. It has to collapse all the way to a singularity for the same reason that you can't keep from moving into the future! Nothing else that goes into that region afterward can avoid it either, at least in this simple case. The event horizon is a point of no return.
In 1971 John Archibald Wheeler named such a thing a black hole, since light could not escape from it. Astronomers have many candidate objects they think are probably black holes, on the basis of several kinds of evidence. Typically these are dark objects whose large mass can be deduced from their gravitational effects on other objects, and which sometimes emit X-rays, presumably by giving a phenomenal acceleration to infalling matter. The properties of black holes that we've talked about here are based on general relativity, which is a theory well supported by available evidence. Nevertheless, they are entirely theoretical, and they don't include probable effects due to quantum mechanics. It's doubtful that "simplistic" black holes with the bizarre properties that are described here really exist, and the physical theories that apply to extreme conditions are very difficult to test and evolve. It has been said that new theories are never really accepted as such; it's just that the old guard dies out, and the new generation doesn't know any better. So it is with black holes. Their existence is nowadays taken for granted by young scientists, but it's important to realise than none have ever conclusively been found.