Off the top of my head, since I'm not at home at the moment, I have (and recommend):
"Bizarre Love Triangle", New Order covered by Frente
I'm Your Fan, an album of covers of Leonard Cohen songs
"'97 Bonnie & Clyde", the Eminem track, redone by Tori Amos... which is, let's just say....iiiinteresting
Various stuff by Me First and the Gimme Gimmes
That erstwhile theme song "Spiderman", covered by the Ramones
"Gone Baby Gone", the Violent Femmes song covered by Gnarls Barkley
"Cactus", the Pixies covered by David Bowie
"Big Mouth Strikes Again", the Smiths, covered in ... I want to say Estonian by a band who's name I can't remember. I dunno. They were on NPR once. It's probably the cover most like the original on this list.
"About a Girl", by Cibo Matto
A little further off the rock/pop beaten track:
I find Weird Al's polka medleys of covers amusing, but I can see how that might not be for everyone.
"Seven Nation Army" covered by Nostalgia 77
"My Favorite Things" by Coltrane, but the Dave Brubeck Quartet version isn't bad, either
"Linus and Lucy" by the Wynton Marsalis Septet
Various Radiohead tracks on piano by Christopher O'Riley
"Shine" (that Collective Soul song) as covered by Dolly Parton, which to me is everything a cover should be -- it's good even if you haven't heard another version and it's distinctively unique, but after hearing it you listen differently to the original, too.
"Rodeohead" (another parody medley) by Hard 'n' Phirm
Covers
Off the top of my head, since I'm not at home at the moment, I have (and recommend):
A little further off the rock/pop beaten track:
-Howard