Theia is the most commonly accepted name for the hypothetical protoplanet that may have collided with the Earth 4.5 billion years ago; the resulting debris coalesced into our moon.
The name Theia was suggested in 2000 by the English geochemist Alex N. Halliday, but the idea of a rogue planet impacting with the Earth to create the moon goes back to Canadian geologist Reginald Aldworth Daly, who suggested it in 1946. Halliday named the planet after the Greek Titan who mothered Selene, the goddess of the Moon. Others have called it, for reasons somewhat unclear, Orpheus.
This sort of theories on the moon's formation are generally called the The Big Whack or The Big Splat, or, in less exciting rhetoric, the Giant Impact hypothesis. The best guess currently is that Theia was about the size of Mars, perhaps one of many such planets wandering around the early solar system. The Earth probably experienced multiple collisions, although we don't have such striking evidence of most of these impacts. It is thought that Theia may have originated in an Earth-Sun Lagrangian point, either L4 or L5.
The collision with Theia would explain some mysteries -- starting with the existence of our overly large moon, of course. It would also help explain why the Earth has such a hot core, as the collision would have created immense friction, why moon rocks are so close in composition to those of Earth, and the high angular momentum of the Earth–Moon system.
There are some problems with this hypothesis, however. For example, it is not clear why these impacts would have created the moon, rather than debris rings. The composition of the moon also matches that of Earth, with no evidence of outside materials in the mix (oxygen isotopic ratios are different for each of the major bodies of the solar system, and would presumably be so for Theia as well). The impact hypothesis would also predict that both the Earth and the moon would have a period of large oceans of molten rock -- which the moon did, but the Earth apparently did not.
Despite these problems the Giant Impact hypothesis is probably our best bet on how the Earth-Moon system formed. There are other possibilities, but most of them do include a Mars-sized planet (or two) crashing into to something. However, in 2012 Robin M. Canup proposed that thinking of the collision as being between Earth and Theia is misleading, and that the initial impact may have been between two larger wandering planets, with the debris reforming into the Earth-Moon system. This theory does not currently appear to have a name to clearly distinguish it from the Giant Impact hypothesis, and the planets involved have not yet been hypothetically named.