Oh ICK. IMHO, of course 
Manfred Mann covered of Blinded by the Light, For You, and Spirit in the Night (all Bruce Springsteen originals) so well that a good many people thought MM’s versions were the originals and were disappointed when they later heard Springsteen’s versions.
Type O Negative’s medley of the Beatles’ “Daytripper/If I Needed Someone.”
Sepultura’s version of New Model Army’s “The Hunt”.
Oh, it’s great! They sound like a heavy metal version of the Doors…that is, even more like a heavy metal version of the Doors than they usually did in those days…complete with quasi-“Light My Fire” extended organ and guitar solos.
Neil Finn does a great cover of this song.
Smashmouth’s “I’m a Believer” is vastly superior to the original. I absolutely hated the Monkees version, but I just love the remake!
Dont forget the cover of NWA’s “Boyz in the Hood” done by Dynamite Hack! Fire up Kazaa you sweaty bitches!
The Negro Problem - “MacArthur Park” (Donna Summer I think)
Jon Brion - “Voices” (Cheap Trick)
hehe…Wesley Willis’ “Girls on Film”…hehehe…
Chantal Kreviazuk…
was the one whose version was from the movie soundtrack… errr… where they went to space… Ben Affleck… um, Bruce Willis… whatever it was called
http://www.chantal-kreviazuk.net/
Some folks have already mentioned Stevie Ray Vaughn’s version of “Little Wing,” but I also think Derek & The Dominos did an excellent version of it.
Bah! You beat me to it. I love that version.
As for Me First and the Gimme Gimmes, well, I’m listening to them right now. Good stuff.
The Damned upped the emotional ante in Love’s “Alone Again Or”.
The Red Hot Chili Peppers speeded up “Love Rollercoaster” by The Ohio Players – but I love both versions. Call it a toss-up.
I prefer Talking Heads’ cover of “Take Me To the River” to Al Green’s original, but I heard the Head’s version first (by many years).
And the definitive version of “All Along the Watchtower” is the 1977 punk cover by XTC. 
It was originally done by Darlene Love. I’m not sure the U2 version is better, but it is good.
And now a few Beatle covers – Art Garfunkle has a great version of “I Will” and James Taylor’s version of “Day Tripper” is good too.
Continuing with Beatles covers:
Stevie Wonder’s “We Can Work It Out” is… well, I’m not sure it’s better, but it’s damned good, and takes the song in a different direction. One that has mad flava.
I think the best version of “Across The Universe” was done by Rufus Wainright on the “I Am Sam” soundtrack.
I don’t think that anyone is fit to judge which version of “Voodoo Chile” is better between Hendrix and Vaughn. I figure that with that, it’s a question that can only be answered in heaven (or wherever you go when you die). It’d be a big ol’ battle of the bands, but then, it would turn into the GREATEST JAM SESSION EVER.
Hole’s version of “Gold Dust Woman” kicks the hell out of Fleetwood Mac’s version. Even if it did have non-rocknroll, Hollywood-style Courtney Love.
Wilson Pickett’s version of “Land of 1000 Dances” – I was convinced that it WAS the original, it was so good. (The original was done by Cannibal & The Headhunters)
I’m kind of torn on “Knock On Wood” – The Eddie Floyd original had that sweet Stax sound, but Amii Stewart’s had one hell of a backbeat…
“Superstar” was made famous by The Carpenters (but I think that Bette Midler sang the original), but Sonic Youth actually made it MORE melancholy.
Gray Matter plays a rockin’ cover of The Beatles’ “I am the Walrus”, If I do say so myself.
Jeff Buckley’s cover of Leonard Cohen’s “Hallelujah” is transcendent beyond words.
I happen to like the warped-ness of Tori Amos’ cover of Eminem’s “'98 Bonnie & Clyde.” Tori also does an incredible cover of Fleetwood Mac’s “Landslide” (which I heard live last Saturday) and The Rolling Stones’ “Angie” that surpass the originals, both of which are spectacular in their own rights.
P.S.: Whoever does that godawful cover of Springsteen’s “Blinded By the Light,” who sounds like he’s saying “wrapped up like a douche,” should be shot.
Bow Wow Wow’s cover of The Strangelove’s I Want Candy
My favorite version is Delbert McClinton’s.
My two favourites are:
Butthole Surfers cover of “Underdog”
Revolting Cocks doing “Do Ya Think I’m Sexy”
Sinead’s All Apologies
UB40’s Can’t Help Falling in Love
Oasis did a superb cover of David Bowie’s “Heroes” that outdid the original. Probably the best cover of any song ever.
And Noel Gallagher did great acoustic versions of Paul Weller’s “To Be Someone” and Slade’s “Merry Xmas Everybody”.