This letter was easier to choose. While I like The Doors they aren't the top of the pile here. One of the weirdest bands I know is, The Decemberists. I love the pop mixed with sea shanties and topped off with just the right amount of references to culture and literature. They are pretty brilliant in my opinion.
It was tough to pick one song but I picked the first one I heard and lead me down the path of owning everything they have made. It is a great antiwar protest song while still being smart and cleaver.
Bonus post: Another good song