Artic Monkeys, Sigur Ros, Radiohead, Muse, Keane, Mumford & Sons, You Me At Six, Zero 7. Bit of something for everyone there.
And surely you don't need to be told about the classics such as The Beatles and Queen...
Also some good Canadian bands (not European, but for some reason the USA and Europe are the only two places in the world?) such as Metric and Arcade Fire are worth a shot.
Then again, since when is Europe the only non-american place where music is made? I implore you to check out some good Australian bands, such as: Crowded House, Eskimo Joe, Gyroscope, Jet, Kisschasy, Little Red, Pete Murray, Powderfinger, The Butterfly Effect, The Temper Trap, Thirsty Merc and Van She. (in alphabetical order of course!)
Also still some great music coming out of the USA of course, if you can only find rap and hip hop you're looking in the wrong places.