Fëa and hröa
In J. R. R. Tolkien's legendarium, fëa and hröa are words for "soul" (or "spirit") and "body". The plural form of fëa is fëar (pronounced [ˈfɛ.ar]) and the plural form of hröa is hröar (pronounced [ˈrɔ.ar]). (The words are also spelt fea and hroa because the spellings ëa, öa, where they occur, are only meant to remind people used to English orthography that the two vowels should not be drawn together in speech as in English "sea" or "load")
The Children of Ilúvatar (Elves and Men) are described as existing as two parts: they have a "spirit" or "soul" called fëa, and a body or hröa which is made out of the matter of Arda (erma); for this reason hröar are Marred (or, using another expression by Tolkien himself, contain a "Melkor ingredient"), and this explains also why it was easier for Melkor and his servants to seduce Men rather than Elves, because mannish fëar have much less control of their hröar. According to the Elves, the fëa is powerless without the hröa, and likewise the hröa would die without the fëa.
The Elves' fate is to live as long as Arda exists; they are bound to the world and cannot leave it. Unlike Men, Elves do not die of disease or of old age. However, Elves may be slain or lose the will to live, for instance because of grief. When an Elf dies, the fëa leaves the hröa, which then "dies". The fëa is summoned to the Halls of Mandos, where it is judged; however as with death their free-will is not taken away, they could refuse the summons. If allowed by Mandos, the fëa may be re-embodied into a new body that is identical to the previous hröa. (In earlier versions of the legendarium it may also re-enter the incarnate world through child-birth.)
Glorfindel died in battle during the First Age, and is the only Elf known to have returned to Middle-earth (probably around 1600 Second Age). This was remarkable not only because Elves after their re-embodiment were unlikely to leave Aman and return to Middle-earth, but also because he was a Noldo under the Doom and Ban of Mandos.
A fëa may decide to stay in the Halls of Mandos, or it may be denied re-embodiment. This would normally only occur if it had done much evil (or those Noldor under the Doom of Mandos). In such a case the fëa might have to wait very long or might never be allowed to leave Mandos. One example of such an Elf is the Noldorin leader Fëanor (whose name, incidentally, means "fire-fëa", that is, "Spirit of Fire").
The situation of Men is different: a Mannish fëa is only a visitor to Arda, and when the hröa dies, the fëa, after a brief stay in Mandos, leaves Arda completely. Originally men could "surrender themselves: die of free will, and even of desire, in estel" but Melkor made Men fear death, instead of accept with joy the Gift of Eru. Men able to surrender themselves were the Númenóreans in their first centuries, and Aragorn.
For this reason it is said the Fates of Men and Elves are sundered. The choice of the Half-elven is one of the fëa, not of the hröa. Lúthien is one of only two Elves to have been permitted to share the Fate of Men (the other being Arwen, whose romance with Aragorn parallels Luthien's), and so her fëa passed out of Arda and was lost to her kin (together with her lover Beren, the only known Man to have been permitted to be re-embodied and return to Middle-earth for a while). Tuor son of Huor is a similar (and for his race, unique) exception; he was permitted a spiritual "conversion" so that his fëa would be counted as one of the Elves.
Not only the Children of Ilúvatar possessed fëar: they were also granted to the Ents and some animals, such as Huan the Hound and perhaps also the great Eagles. When Ilúvatar adopted Aulë's children, the Dwarves, he granted them, too, fëar of their own.