Jeff Buckey - Hallelujah
This is arguably the most beautiful song I have ever heard. Other people have sung the song and it was nice, but Jeff's version is stunningly beautiful. I was making a list of 'masterpieces' about a year ago and this made it. Beyond words, really.

I also agree with Johnny Cash - Hurt and Reel Big Fish - Take On Me

Here's a few others for people to think over... the bold ones in particular I'm personally quite fond of. A lot of them I've just listed because they're incredibly famous rather than I personally like them, though.
Gene Kelly - Singin' in the Rain
Joe Cocker - With a Little Help from My Friends
The Beach Boys - Barbara Ann
Counting Crows and Vanessa Carlton - Big Yellow Taxi
DJ Sammy - The Boys of Summer
David Bowie - China Girl
N-Trance - Do Ya Think I'm Sexy?
Cyndi Lauper - Girls Just Want To Have Fun
Elvis Presley - Hound Dog
The Clash - I Fought The Law
Green Day - I Fought The Law
UB40 with Chrissie Hynde - I Got You Babe
Whitney Houston - I Will Always Love You
Diana Ross - I Will Survive
Mariah Carey & Trey Lorenz - I'll Be There
Smash Mouth - I'm A Believer
Guns N' Roses - Knockin' on Heaven's Door
Pearl Jam - Last Kiss
Eric Clapton - Layla
Club Nouveau - Lean On Me
Guns N' Roses - Live and Let Die
Gary Jules - Mad World
Louis Armstrong - Moon River
Annie Lennox - No More I Love You's
Sinéad O'Connor - Nothing Compares 2 U
Dean Martin - Raindrops Keep Fallin' on My Head
UB40 - Red Red Wine
Aretha Franklin - Respect
Soft Cell - Tainted Love
Sixpence None the Richer - There She Goes
Go West - The Tracks of My Tears
The Righteous Brothers - Unchained Melody
My Chemical Romance and The Used - Under Pressure (for months, I refused to listen to this cover because the original was so great and I didn't think they'd do it any justice - I accidentally heard it some day and admittedly liked it)
Run DMC and Aerosmith - Walk This Way (not sure if this one counts)
Bette Midler - The Wind Beneath My Wings
Dusty Springfield - You've Got a Friend
Quindon Tarver - When Doves Cry

Some of these I was really surprised to discover they were covers, though.