Here is another link for Eowyn's dirge which lyrics differs slightly from the one in Oddwen's link. I don't know which one is more accurate, though.
That link above provides also lyrics and translations for other songs / dialogues heard in the trilogy.
I think Liv sings very beautifully during HoH though at first it felt weird to hear someone singing that kind of a song in English instead of Sindarin or Quenya. Btw, are the credit songs prohibited in this discussion? Anyway, at least one of my very favourites is the elves' song in the Shire.
Click for the soundfile. Here are the lyrics:
A Bereth thar Ennui Aeair!
Calad ammen i reniar
Mi 'aladhremmin ennorath.
A Elbereth Gilthoniel
i chin a thűl lín míriel...
'O Queen beyond the Western Seas!
O light to us that wander
Amid the tree-woven middle-lands.
O Elbereth Star-kindler
the eyes and breath your [are] shining-like-jewels...'
The feeling of the song is quite plaintive but on the other hand very hopeful and pacific. I think it fits perfectly for elves. The melody is mysterious or even a bit eery. It's a shame that Frodo and Sam can't keep their mouths shut during that beautiful piece
It doesn't sound at all like a regular walking song (such as the hobbits sing). At that point in the movie we haven't actually seen many elves (except in the prologue). Elves are presented in that scene as very noble and lofty characters and their song emphasizes the feeling. I like the idea that despite the elves are walking through the Shire, they are just singing their own ancient songs about their favourite Vala ignoring every disturbing creature around them.