Covers that are better than the original

[empath]

"Smooth Criminal" MJ first, then Alien Ant Farm.

Both are good, and are better for different reasons; the decision can lay in simply the mood I'm in at any given time.

[Ben]

"Music"
by Madonna (remixed)
covered by Out of Your Mouth
Sounds kinda shitty on youtube though.

It's debatable if it's better or not
"Any Way You Want It" - Journey covered by Rise Against, I like the lead singer of Rise Against


Edit:
I totally forgot William Shatner's "Has Been" Album with Ben Folds.
Common People
by Pulp
covered by William Shatner & Ben Folds

[InsaneFool]

Covers that I like:

Battery - Machine Head (Originally by Metallica)
Careless Whisper - Seether (Originally by Wham)
Imagine - A Perfect Circle (Originally by John Lennon)
Another Brick IN the Wall - Korn (Originally by Pink Floyd)
Land of Confusion, Shout 2000 - Both by Disturbed (Originally by Genesis, and Tears for Fears respectively)
Rebel Yell - Drowning Pool, and Children of Bodom (Originally by Billy Idol)
F*#! Tha Police, You Spin Me Round (Like a Record) - Dope (Originally by N.W.A. and Dead or Alive respectively)
Holy Diver - Killswitch Engage (Originally by Dio)
Simple Man - Shinedown (Originally by Lynyrd Skynyrd)

Thats all I can think of right now.

[Kara]

A great cover I can think of is only recorded live at all is Elliot Smith's cover of Waterloo Sunset by The Kinks.

A terrible terrible cover that comes to mind is The Scissor Sisters doing a cover of Pink Floyd's Comfortably Numb.

[Dazzla]

I'm going to put forward 'Ugly Kid Joe's rendition of 'Cats in the Cradle'.

[2stepz]

I want you to want me - (covered by Letters to Cleo, original by Cheap Trick)
Because of you (Kelly Clarkson, covered as a Clarkson/Reba McEntire duet)

[What?]

I'm probably about to commit musical blasphemy, but I find myself preferring Limp Bizkit's version of "Behind Blue Eyes" over the original. I just find their vocals a lot clearer. I'm also apparently the only person on earth whose favorite rendition of Hallelujah is Rufus Wainwright's.

[Mad Madam Mimm]

I'm going to put forward 'Ugly Kid Joe's rendition of 'Cats in the Cradle'.

Ditto that. I love Ugly Kid Joe, their greatest hits album is hilarious on a lazy summer day.

There was an album made by George Martin, the producer of the Beatles. It was basically Beatles songs coverred by celebrities who added their own spin on the songs, and they were all really interesting to listen to.
All songs by John Lennon and Paul McCartney, except where noted. My favourites were-

"Come Together" Featuring Robin Williams and Bobby McFerrin on vocals
"A Hard Day's Night" Featuring Goldie Hawn on keyboards and vocals
"I Am the Walrus" Featuring Jim Carrey on keyboards and vocals
"Here Comes the Sun" Featuring guitarist John Williams
"Being for the Benefit of Mr. Kite" Featuring Billy Connolly on vocals
And
"In My Life" – 2:29 Featuring Sean Connery on lead spoken word vocal

It was interesting because they were so different to the originals. Like Hard Day's Night was a slow, blues/jazz number, which was really well done.

[Zhyard]

These are the ones I could think of at the top of my head.

Hurt by Johny Cash (NIN), as many others have allready mentioned.
Das Modell by Rammstein (Kraftwerk).
Umbrella by Lillasyster (Rihanna).
Tainted Love by Marilyn Manson (Gloria Jones).

[2stepz]

Any Eagles fans?

I know it's blasphemy for an Eagles fan to enjoy the album, but I enjoyed "Common Thread" which was an album of Eagles songs covered by country artists.

Clint Black did possibly the best rendition of "Desperado" I've ever heard, Travis Tritt's cover of "Take It Easy" reached the top of the charts, and the four songs covered by the ladies (Trisha Yearwood, Suzy Bogguss, Lorrie Morgan, and Tanya Tucker) put a different spin on some traditionally male oriented songs. I only wish I could compliment John Andersons' cover of "Heartache Tonight."

Different album, but Ricochet did a acapella cover of "Seven Bridges Road" that is still on my list of favorite recordings. Their voices blend so perfectly for that song...

[GreigKM]

Please Mr.Postman - The Beatles
Brown Eyed Girl - Everclear
Boys - The Beatles
Twist and Shout - The Beatles
And many others.