I don't have a favorite "Alt Rock Band" but i like Foster the People and Alt J (a lot) my favorite song from foster the people is call "Goats In Tree" and "Fire Escape".
"Alternative rock" is a weird label for a mish-mash of a bunch of different styles. It's a really loose term, particularly these days when there a probably hundreds of rock subgenres.
The bands I associate most with the label are The Smashing Pumpkins and Pearl Jam - and basically all of that post-Nirvana grunge stuff. My favourite artist that I've seen categorised as alt. rock's definitely Jeff Buckley, though.