I have a bunch of favorites, but here's a few off the top of my head by some of my favorite artists:
Smashing Pumpkins - Landslide (Fleetwood Mac), Cinnamon Girl (Neil Young), Rock On (David Essex), Don't You Want Me Baby (Human League), Transmission (Joy Division), Number of the Beast (Zwan cover of Iron Maiden song), I'm Eighteen (Zwan cover of Alice Cooper song), To Love Somebody (Corgan covering Bee Gees)
Radiohead - Nobody Does it Better (Carly Simon), Wish You Were Here (Yorke w/ Sparklehorse) (Pink Floyd), I'll Wear it Proudly (Costello), After the Gold Rush (Neil Young), The Headmaster Ritual (The Smiths)
Pearl Jam - Fuckin' Up (Neil Young), Masters of War (Dylan), Last Kiss (Wayne Cochran), 25 Minutes to Go (Johnny Cash)
Wilco - I Shall Be Released (the Band), John Wesley Harding (Dylan), Comment (If All Men Are Truly Brothers) (Charles Wright), Any Major Dude Will Tell You (Steely Dan), Love and Mercy (Brian Wilson), Thirteen (Big Star)
The Flaming Lips - Can't Get it Out of My Head (Minogue), After the Gold Rush (Neil Young), Knives Out (Radiohead), The Golden Age (Beck), Bohemian Rhapsody (Queen), Death Valley '69 (Sonic Youth), Life on Mars (Bowie)
Ben Folds - Tiny Dancer (Elton John), Them that Got (Ray Charles), Bitches Ain't Shit (Dr. Dre), Race for the Prize (Flaming Lips)
Counting Crows - Carmelita (Warren Zevon), A Good Year for the Roses (Elvis Costello), Blues Run the Game (Jackson Frank), Wiseblood (Patrick Winningham Band)
Dave Matthews Band - All Along the Watchtower (the Red Rocks '95 version is especially good), If I had Boat (Lyle Lovett), The Maker (Daniel Lanois), Sweet Caroline (Neil Diamond)
Marilyn Manson - Highway to Hell (AC/DC), Sick City (Charles Manson), Down in the Park (Gary Numan), Get My Rocks Off (Dr. Hook), Suicide is Painless (Mandel and Altman)
Nirvana - Ain't it a Shame (Ledbetter), Grey Goose (Ledbetter), They Hung Him of a Cross (Ledbetter), Where did you sleep last Night? (Ledbetter), D-7 (Wipers), Return of the Rat (Wipers), Lake of Fire (Meat Puppets), The Man Who Sold the World (Bowie)
Tom Waits - Tom Trauberts' Blues (parts of "Waltzing Matilda" in it), Goodnight Irene (Ledbelly), The Return of Jackie and Judy (The Ramones), Young at Heart (popularized by Sinatra) - others from the amazing Orphans box set and more which escape me at the moment...
Here's some other good ones I have in my collection:
Beck - Do You Realize? (Flaming Lips)
Polyphonic Spree - Lithium (Nirvana)
Vanessa Carlton - Paint it Black (Rolling Stones)
Tori Amos - Smells Like Teen Spirit (Nirvana)
Susanna and the Magical Orchestra - Love Will Tear Us Apart (joy Division)
The Eagles of Death Metal - Stuck in the Metal (cover of Stealers Wheel "Stuck in the Middle")
Gypsy Kings - Hotel California (Eagles)
Metallica - Turn the Page (Bob Seger)
Oasis - Heroes (Bowie) and Cum on Feel the Noize (Quiet Riot)
The White Stripes - Walking with A Ghost (Tegan and Sara)
There's also some excellent albums of covers I'd recommend:
Springsteen - We Shall Overcome (the Seeger sessions)
Johnny Cash - he has a whole series of cover albums (American Recordings), including Hurt by NIN (as mentioned) and a nice cover of Rowboat by Beck, among many many others.
Billy Bragg and Wilco - Mermaid Ave Vol 1 and 2 (Guthrie lyrics put to music, not really "covers" I guess)
Eric Clapton - Me and Mr. Johnston
Nels Cline and Gregg Bendian - Interstellar Space Revisited: The Music of John Coltrane
Bobh Dylan - Self Portrait (mostly covers) and Good as I Been to You. There's also a 9 volume set of Dylan performing covers out there called the "1988-2000 Neverending Tours Covers Collection"
A Perfect Circle - eMotive
Encomium - A Tribute to Led Zeppelin (lots of well known 90s bands covering Zeppelin tunes)
Also, everyone who is a fan of covers should check out that "other" cover project site:
http://www.coversproject.com/