Elvish

There were two languages of the Eldar, Quenya and Sindarin. Quenya was the language of the High-elves and Sindarin was the language of the Grey-elves. The two had originally had many similarities, but they had grown far apart by the Third Age.

Quenya

Quenya was the language of the High-elves. By the Third Age it was a "dead tongue" and was used only for ceremony, lore, and song. (RotK 468) It was heavily influenced by Finnish. (Carpenter 101)

Sindarin

Sindarin was the language of the Grey-elves of Doriath. It had been adopted for daily use by the Elves of Middle-earth. (RotK 468) It was modelled on Welsh. (Carpenter 101)

Here is an example of Quenya:

A Olorin i yaresse mentaner i numeherui

Tirien i romenori

maiaron i oiosaila

manan elve eteuanne

Nuryie i melanelve?

Olorin who once was

Sent by the Lord of the West

to guard the lands of the East

wisest of all Maiar

what drove you to leave

that which you loved? (Boyens 1)

More Quenya:

A star shines at the hour of our meeting

Elen sila lumenn omentilmo

Good day

Quel re

Good morning

Quel amrun

Good night

Quel undome

Greetings

Verdui'

My heart sings to see thee

Cormamin lindua ele lle

Pleasure meeting you

Saesa omentien lle

Namarië

Farewell (Schultz 1)

(Boyens Philippa, and Howard Shore. "Lament for Gandalf." New Line Tunes, 2002.)

(Carpenter, Humphrey. J.R.R. Tolkien: A Biography. Boston, New York: Houghton Mifflin, 1977.)

(Schultz, Monica. "Learn Elvish!" Y.E.S. (Young Eagles Society) April/May/June 2004: 14)

(Tolkien, J.R.R. The Return of the King. New York: Ballantine Books, 1956.)

Etcetera, etcetera, etcetera!