Normally, I hate covers and just wonder why most people bother, as what tends to come out is a karaoke style mish-mash of the original.
But then there are also the literally hundreds of amazing covers where bands or individuals have stamped their own personalities on a song and changed it to suit themselves - thus putting some creative, imaginative input into the process.
Where to start...
There are the obvious ones, like just about every cover The Beatles did on their first albums (as they were contractually obliged to do), every cover Metallica did (apart from their Motorhead ones), Motorhead's many, many amazing covers including Johnny Burnette's "Train Kept a-Rollin", the Pink Fairies "City Kids" and Iron Maiden's "The Trooper"...
The one to hear is an entire album;
Easy Star All-Stars "Dub Side of the Moon".
The follow-up "Dub Computer" (credited to Radiodread) is equally interesting;
Hayseed Dixie, who specialise in country versions of AC/DC are very cool too;
/edit:
The legendary William Shatner has done so many that are awesome, like "Lucy in the Sky With Diamonds", and Pulp's "Common People", but my favourite is the (possibly unintentionally) hilarious version of Elton John's "Rocket Man";
Some you should also know about if only to avoid;
ANY cover of ANY Sex Pistols song. People just don't get it.
Dream Theater's covers. They crucified "Master of Puppets", and you just don't want to know about "Number of the Beast" and "Dark Side of the Moon", trust me.
Wet Wet Wet's atrocious cover of The Troggs' "Love is all Around".
Rolf Harris' cover of "Bohemian Rhapsody".