Author J. R. R. Tolkien created many languages for his Elves, which eventuated in the creation of a mythology (expounded in his books), complete with races, to speak the tongues he had constructed. His interest was primarily philological, and he said his stories grew out of his languages. The languages were the first thing Tolkien created for his mythos, starting with what he originally called "Qenya", the first primitive form of Elvish. This was later called Quenya (High-elven) and is one of the two most complete of Tolkien's languages (the other being Sindarin, or Grey-elven). The phonology, vocabulary and grammar of Quenya and Sindarin are strongly influenced by Finnish and Welsh, respectively. In addition to these two, he also created several other (partially derived) languages. In addition to Tolkien's original lexicon, many fans have contributed words and phrases, attempting to create a language that was fully usable in reality.
Other Elvish languages
Since Tolkien, others have invented Elvish languages in their own fiction. Several borrow sounds and forms (or even whole words) from Tolkien's Elvish languages, especially Quenya, while others are quite distinct.
Elven, The language of RuneScape's Elves, is basically Welsh with a slight twist.
The Ancient Language The language of the elves in Eragon. Also used by the riders and other magic users to cast spells. Was the language of the now extinct Grey Folk. One cannot lie in the Ancient Language and is bound by what one says in it.
Elven, the language of the surface Elves of Dungeons & Dragons (of which there are several dialects including Ssamath of the Drow, which drow use in conjunction with Deep Speech).
Elvish, the language of the Dalish elves in the Dragon Age series of videogames (and, to an extremely limited extent, the City Elves as well).
^Solopova, Elizabeth (2009), Languages, Myths and History: An Introduction to the Linguistic and Literary Background of J.R.R. Tolkien's Fiction, New York City: North Landing Books, "Invented Languages", ISBN 0-9816607-1-1