Like black holes, wormholes arise as valid solutions to the equations of Albert Einstein's General Theory of Relativity, and, like black holes, the phrase was coined (in 1957) by the American physicist John Wheeler. Also like black holes, they have never been observed directly, but they crop up so readily in theory that some physicists are encouraged to think that real counterparts may eventually be found or fabricated.
In 1916, the Austrian physicist Ludwig Flamm, while looking over Karl Schwarzschild's solution to Einstein's field equations, which describes a particular form of black hole known as a Schwarzschild black hole, noticed that another solution was also possible, which described a phenomenon which later came to be known as a “white hole”. A white hole is the theoretical time reversal of a black hole and, while a black hole acts as a vacuum, drawing in any matter that crosses the event horizon, a white hole acts as a source that ejects matter from its event horizon. Some have even speculated that there is a white hole on the "other side" of all black holes, where all the matter the black hole sucks up is blown out in some alternative universe, and even that what we think of as the Big Bang might in fact have been the result of just such a phenomenon.
Flamm also noticed that the two solutions, describing two different regions of space-time could be mathematically connected by a kind of space-time conduit, and that, in theory at least, the black hole "entrance" and white hole "exit" could be in totally different parts of the same universe or even in different universes! Einstein himself explored these ideas further in 1935, along with Nathan Rosen, and the two achieved a solution known as an Einstein-Rosen bridge (also known as a Lorentzian wormhole or a Schwarzschild wormhole).
To better visualize a wormhole, consider the analogy of a piece of paper with two pencil marks drawn on it (to represent two points in space-time), the line between them showing the distance from one point to the other in normal space-time. If the paper is now bent and folded over almost double (the equivalent of drastically warping space-time), then poking the pencil through the paper provides a much shorter way of linking the two points, a short-cut through space-time much like a wormhole.
Some theorists are encouraged to think that real counterparts may eventually be found or fabricated and, perhaps, used as a tunnel or short-cut for high-speed space travel between distant points or even for time travel (with all the potential paradoxes that might entail). However, a generally accepted property of wormholes is that they are inherently highly unstable and would probably collapse in a much shorter time than it would take to get through to the other side. At any rate, it is predicted that they would collapse instantly if even the tiniest amount of matter (even a single photon) attempted to pass through them.
Although some possible theoretical ways around this problem have been suggested (for example, using “cosmic strings” or “negative matter” or some other exotic matter with “negative energy”) to prevent the wormhole from pinching closed, the idea remains largely in the realm of science fiction for the time being. It has, however, still not been mathematically proven beyond all doubt that some kind of exotic matter with negative energy density is an absolute requirement for wormholes, nor has it been established that such exotic matter cannot exist, so the possibility of a practical application of the theory still remains.
Because a wormholes is a conduit through 4-dimensional space-time, and not just through space, Stephen Hawking and others have also posited that wormholes might theoretically be utilized for travel through time as well as through space, although it is widely believed that time travel into the past will never be possible due to the potential for paradoxes and self-destructive feedback loops.