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).
The Elvish language(s) of the Warcraft universe (Darnassian, Nazja, and Thalassian), are not described in detail but superficially resemble Tolkien's. Together, Darnassian, Nazja, and Thalassian are considered the modern elvish tongues of the Warcraft universe spoken by the modern Kaldorei, the Naga, and the highborne (respectively) while Elvish itself is considered an ancient tongue that is no longer used as a primary language. It is assumed that Elvish is the language from which Darnassian evolved; Darnassian then branched into Nazja, spoken underwater by the kaldorei that followed Queen Azshara after the sundering, and later on, Thalassian, which is spoken by the highborne as well as the Blood Elves.
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).
The Elves, or Mer, of the universe of The Elder Scrolls, use various languages derived from ancient Ehlnofex. Such languages include Dunmeris, Pyandonean, Orcish (Orsimeris) and Bosmeris, among others.
^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