Famous Blue Raincoat by Jennifer Warnes is one of the best cover albums.
Oh, yes, I agree.
Clive Gregson and Christine Collister put out “Love Is A Strange Hotel”, “low-key acoustic album collection of covers, inlcuding Aztec Camera’s ‘How Men Are’ and 10cc’s ‘Things We Do for Love’.” It’s one of my favourite records of theirs, though Gregson is a very good songwriter on his own.
Oh, I forgot: Nanci Griffith, Other Voices, Other Rooms and **Other Voices, Too (A Trip Back to Bountiful). **
The Beatles “Sgt. Pepper’s” album was notoriously covered by the BeeGees, who had enjoyed enormous success in the late 70’s with their own self-penned songs. Of course, there were other contributors to the album, including Steve Martin, Aerosmith, George Burns and Alice Cooper. 
I confess that I really like Aerosmith’s version of “Come Together” and Sandy Farina’s version of “Strawberry Fields.”
Richard Thompson’s 1000 Years of Popular Music is a great example of this. He covers such diverse artists as Orazio Vecchi, Gilbert and Sullivan, Hank Cochran, and Britney Spears.
Besides The Covers Record mentioned by Monkey Chews, Cat Power’s most recent CD was mostly covers - Jukebox.
Bryan Ferry - Dylanesque. I like it, but it’s no match for his A Hard Rain’s A-Gonna Fall.
Both of those CDs are fun and “Black Hole Sun” and “Smells Like Teen Spirit” are particularly great.
Richard Cheese does almost nothing but covers of songs in a lounge music style. His cover of Disturbed’s “Down With the Sickness” was in the movie Dawn of the Dead.
Rage Against the Machine also released a CD of covers (their last CD before breaking up actually) called Renegades.
Brazilian superstart Caetano Veloso did an all-English album of covers called A Foreigh Sound, covering Nirvana, Dylan, The Talking Heads, etc., plus doing Cole Porter, George Gershwin, and other songwriters of note.
Rufus Wainwright has a concert album of his recreation of Judy Garland’s 1961 concert at Carnegie Hall. It’s awesome.
The majority of the tracks on Johnny Cash’s American series of albums are covers of other artists’ work, mixed together with newer versions of songs Cash recorded earlier in his career and a very small amount of new material.
His version of Hurt is, as far as I’m concerned, one of the few covers that far outshines the original… as does his version of Personal Jesus.
Love your username 
I give major props to Johnny Cash for the American series albums. His version of “Hurt” made me cry when I first heard it, and moreso when I saw the video that was filmed right before June died. His version of “Personal Jesus” is excellent as well.
My contribution is Def Leppard’s “YEAH!” as well as “Cybernauts Live/Further Adventures of the Cybernauts” which was a side project with Joe Elliott and Phil Collen from Leppard, plus the surviving members of the Spiders From Mars. Their other guitar player, Vivian Campbell, put out an album of blues covers called “Two Sides of If” a couple years back.
Well, there was that lamentable heavy metal album by Pat Boone a few years ago.
I hadn’t thought of Sting’s Songs from the Labyrinth as cover tunes, but I guess it is.
Tom Jones - Reload
And there was also Снова в СССР (Back in the USSR) realeased around 1988. Some favorite 50’s hits compiled primarily for the Soviet Union. Here’s the wiki link:
I considered that, but is Boone primarily a songwriter? Basically all his songs have been covers, so he doesn’t exactly fit in the O.P.'s parameters.
And what’s lamentable about it? I like it. I played it in rotation with my Paul Anka cover albums.
Nena recently released a double-disc of covers, and it’s pretty damn good. Hearing her cover Pink Floyd was surreal.
http://en.wikipedia.org/wiki/Cover_Me_(Nena_album)
fix broken link
*and add home entry with uber-adorable picks of her 
*
Cyndi Lauper released At Last in 2003.
That was a good album – she’s underrated as a singer, IMHO.
Miek en Roel have covered several songwriters in Dutch, including an album of Tom Paxton