Some covers are as good as if not better than the originals.  Some examples, with the original first.  Thanks to our old friend 'williamofockham' for uploading the Morissette number as well as for suggesting the McLachlan and Simon covers.

Beatles, Blackbird.  Sarah McLachlan cover.  Carly Simon cover.

Sam Cooke, Wonderful World.  Simon-Garfunkel-Taylor cover. 

Bob Dylan, Blowing' in the Wind.  Alanis Morissette cover.

Jimmy Jones, Handy Man.  James Taylor cover.

Chantays, Pipeline.  Dick Dale and Stevie Ray Vaughan cover.  Blows the original clean out of the water!

Shirelles, Will You Still Love Me Tomorrow?  This version is untouchable, though covered by many, including Carole King, its composer (with Gerry Goffin).

Chiffons, One Fine Day.  Another untouchable oldie from 1963 penned by Goffin-King.  Carole King does a better job with this one.