This gets asked about a lot ("how can I learn elven"). There is no single or complete language.

• Tolkien invented two (mostly based on welsh and finnish), namely Quenya and Sindarin.
• Silver Elves have fragments
• HomePage The Crisses? has KellaElvish a small project to recreate one elvish language?
• links to various sites, etc.

Assorted links:

• http://members.cox.net/taramiluiel/sindarin_phrases.htm Tara's Sindarin Phrasebook
• http://www.uib.no/People/hnohf/qcourse.htm Quenya course
• http://bwr.hyperchat.com/newchat/u/jessadriel/~bwr/SindarinOnline.html Sidarin resource list