Richard F. Holman is a professor of physics at Carnegie Mellon University. He offers this response:
Wormholes are solutions to the Einstein field equations for gravity that act as "tunnels," connecting points in space-time in such a way that the trip between the points through the wormhole could take much less time than the trip through normal space.
The first wormhole-like solutions were found by studying the mathematical solution for black holes. There it was found that the solution lent itself to an extension whose geometric interpretation was that of two copies of the black hole geometry connected by a "throat" (known as an Einstein-Rosen bridge). The throat is a dynamical object attached to the two holes that pinches off extremely quickly into a narrow link between them.
Theorists have since found other wormhole solutions; these solutions connect various types of geometry on either mouth of the wormhole. One amazing aspect of wormholes is that because they can behave as "shortcuts" in space-time, they must allow for backwards time travel! This property goes back to the usual statement that if one could travel faster than light, that would imply that we could communicate with the past.
Needless to say, this possibility is a disturbing one; time travel would allow for a variety of paradoxical situations, such as going back into the past and killing your grandfather before your father was born (the grandfather paradox). The question now arises of whether it would be possible to actually construct a wormhole and move it around in such a way that it would become a usable time machine.
Wormhole geometries are inherently unstable. The only material that can be used to stabilize them against pinching off is material having negative energy density, at least in some reference frame. No classical matter can do this, but it is possible that quantum fluctuations in various fields might be able to.
Stephen Hawking conjectured that while wormholes might be created, they cannot be used for time travel; even with exotic matter stabilizing the wormhole against its own instabilities, he argued, inserting a particle into it will destabilize it quickly enough to prevent its use. This is known as the Chronology Protection Conjecture.
Wormholes are great theoretical fun, and are seemingly valid solutions of the Einstein equations. There is, however, no experimental evidence for them. This should not stop any budding science-fiction writers from using them as needed!
William A. Hiscock is a professor of physics at Montana State University, Bozeman, and is the director of the Montana Space Grant Consortium. He adds some details:
A wormhole is a tunnel-like connection through space-time, much like the real tunnels bored by worms in a (Newtonian) apple. At present, space-time wormholes are only theoretical constructs derived from general relativity; there is no experimental evidence for their existence. Nevertheless, theoretical physicists study the mathematical properties of space-times containing wormholes because of their unusual properties. Study of such strange geometries can help better distinguish the boundaries of behavior permitted in the theory of general relativity, and also possibly provide insights into effects related to quantum gravity.
A wormhole has two mouths connected by a "throat," and provides a path that a traveler could follow to a distant point. The path through the wormhole is topologically distinct from other routes one could follow to the same destination.
What is meant by topologically distinct? If an ant wished to crawl from one side of an apple to another, there are many possible paths on the surface connecting the starting point to the destination. These paths are not distinct topologically: a piece of elastic string fixed at the starting and ending points, and lying along one such path, could be slid and stretched over the surface to lie along any other such path. Now imagine that the ant instead crawls through a wormhole in the apple. A piece of string passing through the wormhole cannot be smoothly moved in such a way as to lie along one of the surface paths (or through another wormhole with the same end points but different route).