There are oh so many options, that I can scarcely bring myself to choose. So, I'm pretty much just going to come up with the first ones I can come up with that are worthy of mention. It took much consideration to determine the worst, because there is such a large candidate pool. I specifically debated including many of the covers of Personal Jesus (there's a cover that was done by Johnny Cash for fuck's sake!), Rebel Yell, and a recent rock cover of the song Careless Whisper by Seether (oh boy, was that hard to turn down). But, I think I am going to have consider Stahlhammer's cover of Pink Floyd's "Another Brick in the Wall" to be the greatest crime against good music that I could manage to find.
Just so that you could have a reward for your eardrums, and in order to fully appreciate how horrible that cover was, here is the original.
For the best cover, it is a tough call. White Zombie did an excellent cover of "I'm Your Boogieman" but by completely changing the tone of the song and repurposing too much for me to be able to say that it was actually an improvement. The Ataris did an excellent cover of Don Henley's Boys of Summer, but did not improve it by that much and merely capitalized on what was a rather good song to begin with. Korn did a cover of Creep that was essentially as good as the Radiohead original. But, I've got to go with a pet favorite of mine: Gary Jules' cover of the Tears for Fears song "Mad World". The only difference is one of tone, pacing, and sound effects, and both are very good. But, I think that the cover is an incredibly effective alteration of the original to achieve a similar purpose.
And the cover version:
Well, that's that, I suppose. I tag you: mac, Pliny, Mandar, and Maze Monster. Have fun. Over and out.
Edit: Changed version of the original "Another Brick in the Wall" video to one that has less filler.