The most famous and influential Elvish language is Quenya, created by J.R.R. Tolkien for his fictional world of Middle-earth. Tolkien put a tremendous amount of effort into creating Quenya, giving it a rich history, grammar, and vocabulary. He even developed other Elvish languages like Sindarin, which were spoken by different Elven groups.
There are also other constructed Elvish languages based on Tolkien's work, as well as those created for other fantasy settings, like High Elven from Dungeons & Dragons, or Ithkuil, a language created by John Quijada with an incredibly complex grammar.
So, while no "real" Elvish language exists, there are many fascinating and meticulously crafted constructed languages that bring Elvish to life for fans of fantasy literature and games.