Stuff that has not been mentioned:
DeVotchKa - Venus in Furs (Velvet Underground) ... if you don't go find this and listen the hell out of it, you may die.
The Sundays - Wild Horses (Rolling Stones)
Fanfare Ciocarlia - Born to Be Wild (Mars Bonfire)
Melanie - Lay Lady Lay (Bob Dylan)
Robert Plant - Morning Dew (Bonnie Dobson)
Pretty Balanced - Chemo Limo (Regina Spektor)
Billy Corgan ft. Robert Smith - To Love Somebody (the Bee Gees) ... Janis Joplin's cover of this is also very good.
Amanda Palmer - I Will Follow You Into The Dark (Death Cab For Cutie) ... and most other covers this woman has ever performed, including Radiohead's Exit Music (For A Film) and that cover of REM's Everybody Hurts she did with Cormac Bride.
the Dresden Dolls - Blueprint (Fugazi); Will You Smile Again? (...And You Will Know Us By The Trail Of Dead); Two Headed Boy (Neutral Milk Hotel); Rid Of Me (PJ Harvey); New Years Day (U2); Add It Up (Violent Femmes) etc etc etc.
the Smashing Pumpkins - Never Let Me Down Again (Depeche Mode) and Dancing in the Moonlight (Thin Lizzy) ...both better than the originals.
Hard n' Phirm - Rodeohead (Radiohead medley) ... WARNING: if you like your computer screen, do not consume liquids while listening to this.
Gliss - Rhinceros (the Smashing Pumpkins)
Cat Power - I Found A Reason (Velvet Underground)
Lightspeed Champion - Buddy Holly (Weezer)
Weezer - Velouria (Pixies)
the Byrds - Mr. Tambourine Man (Bob Dylan)
A Perfect Circle - most of the album eMOTIVe
Iron & Wine - Such Great Heights (The Postal Service)
the Doors - Alabama Song (Whiskey Bar) (Brecht/Weill)
Get Cape. Wear Cape. Fly. - D.A.N.C.E. (Justice) ...he gets points just for trying.
New Order - Ceremony (Joy Division) ...you may argue this distinction.
....and of course this needs to be seen by everybody:
http://www.youtube.com/watch?v=0WWuZRd-O3cI really want to hear that US Maple cover of Lay Lady Lay now.