I listen to a great amount of what I would term "weird" music (and my dad would term "insanity", even though he often likes it). I do listen to quite a bit of jazz and classical (a topic for another post), but most of my day-to-day listening is weird music.
Central to the "weird" label is my favorite genre of music, New Weird America (or Freak-folk). NWA is a pretty inclusive genre that combines American folk music with psychedelic lyrics (and sometimes music). I fell in love with folk music in early high school through the likes of Bob Dylan and Neil Young (both of whom I still listen to regularly). Then I started getting into all sorts of lyrical and poetic nonsense, and developed an appetite for interesting lyrics in music. Dylan's later work was influential in this, although now I much prefer his earlier stuff now. What is the perfect combination of folk music and awesome, sometimes nonsense lyrics? New Weird America.
Some of my favorites, in no particular order are:
Iron and Wine
She & Him
Six Organs of Admittance
There are plenty of others, and some of these might not really belong to the NWA tag.
But don't think that NWA is the only weird music I listen to. I also like mewithoutYou. mwY is really the only band I like from "post-hardcore" or other such nonsense, because I love their lyrics and their music is beautiful and not oppressive like other hardcore stuff can be.
I also love David Bazan and all his various bands and projects (Pedro the Lion, Headphones, solo stuff). I was tempted to include him in the NWA, but I feel he deserves his own place because of his creativity and the huge degree to which I love his stuff.
There are plenty of other bands that I have forgotten. If you really want to know more about my music preferences, check out my last.fm page.