I always really liked The White Stripes' version of 'Jolene' (original: Dolly Parton), especially the live cut.
My recent favorite cover is Mates of State covering Phantom Planet's 'California' (yes, that song. But it's very un-OCish).
Other honorable mentions:
Wonderwall by Ryan Adams (original: Oasis)
Secret Heart by Feist (original: Ron Sexsmith)
We Will Become Silhouettes by The Shins (original: Postal Service)
In the worst category, too many to enumerate, but 'Guns of Brixton' covered by Nouvelle Vague is just godawful. And Razorlight doing 'Hey Ya!'? Ouch. It was bad when Outkast did it, no need to reinforce that.