There are two types of people in this world.  Those who think Johnny Cash's version of Hurt is better than Nine Inch Nails' and those who deserve to be round up and shot.

Ok, a little extreme but the NIN version is way better.

On topic.

Nirvana- Man Who Sold The World (David Bowie Cover)

Iron and Wine- Such Great Heights (Postal Service Cover)

Sonic Youth- Superstar (Carpenters Cover)

Nirvana- Lake of Fire (Meat Puppets Cover)

"Hurt" by Johnny Cash > Nine Inch Nails version
"Sweet Dreams (Are Made Of This)" by Marilyn Manson > The Eurythmics version
"How I Could Just Kill A Man" by Rage Against the Machine > Cypress Hill version

@bold:  I respectfully disagree.  :)

I find that "The Boys of Summer" by The Ataris is better than the original version by Eagles Drummer/Vocalist Don Henley.

I'm already pissed off.

Cinnamon Girl by Type O Negative:

Originally by Neil Young:

Gamma Ray's cover of Pet Shop Boys' It's A Sin is pretty damn good...

And I prefer Blind Guardian's version of Judas Priest's classic Beyond The Realms Of Death... Hansi sings his fucking balls off here...

Joe Strummer and Johnny Cash - Redemption Song (Originally Bob Marley I think)

http://www.youtube.com/watch?v=cz8_H5m6ymA

Better -

All Along the Watchtower
Landslide - Smashing Pumpkins (like it a bit better than the original Fleetwood version)
Mad World - Like Tears for Fears but not their version
Nirvana covers of the Vaselines
Jolene - White Stripes
Galaxy 500 - Don't let your youth go to waste (Jonathan Richman's version is good, G500 is amazing). Also Listen Outside the Leaves are Falling - better than Lennon/Ono version.
Superman - REM better than the Clique

On Par -

These are ones that are different enough to stand alone but I love both versions

Hurt - NIN = pain of drug addiction. Cash = pain of aging and losing his wife - both amazing
Nirvana covers of Meat Puppets and Bowie
Just Like Heaven - Dinosaur Jr. cover of a Cure classic is so different and great
Sweet Jane - Cowboy Junkies on par with VU

One that pisses me off most when others inevitably cite it - Hallelujah buy anyone other than Leonard Cohen. There are a million good covers but the original is perfect.