Do you know what I like? Hollywood gossip. Do you know what I also like? Bollywood gossip. After that, I would say I also really like covers of songs I know.
While a fair amount has been written on what makes a good cover (see, for example, here and here), the true test for me is whether the song is simply believable coming from the covering artist. Some easy examples of covers that fail this test: H.R.M. Madonna's "American Pie" (what did Don McLean ever do to Madonna?), Macy Gray's "Walk This Way" (of course, I don't believe Macy Gray even when she sings her own songs), and Save Ferris's "Come on Eileen" (a particularly egregious act given the greatness of the original). For a good compilation of some of the worst covers of all time (with embedded videos to make life easier), see here.
Contrast those bad covers with, say, Stevie Wonder's "We Can Work It Out," Sinead O'Connor's version of "Nothing Compares 2 U," Alien Ant Farm's "Smooth Criminal," or The Bangles' take on "Hazy Shade of Winter." A list of some of the best covers (which shares some entries that are on the list of worst covers) can be found here.
An example of a pretty enjoyable cover that has been floating around the internet for the last year or so is Mat Weddle (of Scottsdale, Arizona folk/pop group Obadiah Parker) singing his version of Outkast's "Hey Ya!" This performance, recorded at an open-mic night in Scottsdale, was put in heavy rotation by Arizona radio stations and then gained a lot of popularity via Youtube. It is now available on iTunes.