Fëa and Hröa, fëar and hröar in plural forms, are Quenya words for "soul" (or "spirit") and "body".
The Children of Ilúvatar are described as existing in two parts: they have a spirit or soul (fëa) which comes from the Secret Fire of Ilúvatar, and a body (hröa) which is made out of Arda. According to the Elves, the fëa is powerless without the hröa, and likewise the latter would die without the former.
The fate of the Elves is to live as long as Arda exists as 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 killed by violence or lose the will to live, most of the times because of grief. When an Elf dies, the fëa leaves the hröa, which then dies. The fëa is called to the Halls of Mandos, where it is judged. If allowed by Mandos, the fëa may be reincarnated into a new-born body that is identical to the previous one. The spirit may decide to stay in Mandos, or it may be denied reincarnation, for example if it had done much evil. In such a case the fëa might have to wait very long or might never be allowed to leave Mandos.
To Men however, the situation is different: a mannish fëa is only a visitor to Arda, and when the hröa dies, the fëa temporarily enters the Halls of Waiting before it leaves Arda completely. 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 chose the Fate of Men, and so her fëa passed out of Arda and was lost to her kin.
The Children of Ilúvatar were not the only ones in possession of fëa, the latter were also granted to the Ents and animals, such as Huan and the Great Eagles. When Ilúvatar adopted Aulë's children, the Dwarves, as his own, he granted them, too, fëa of their own. However it is unknown what happens to the fëa of Ents, Animals and Dwarves when they die.
Ainur are, in contrast, spiritual beings that create their own physical bodies to stay in Middle-earth, and are more than capable of changing their hröa to any form as they please. However, their hröa are susceptible to harm and injury, no matter how powerful, even to the point of being completely destroyed. While an Ainu may eventually reform their hröa, their power becomes severely weakened, and such loss of power may rob them of the ability to shapeshift, and even to take any physical form at all, as seen with Sauron.