Jazz
First thing that has to be said about jazz: there are lots of different styles of Jazz. I'll go through them in (roughly) chronological order.
Dixieland & The Big Band Era:
Duke Ellington: Quite a famous name - listen to anything you can find on him if you can. Some key tracks: Take the 'A' Train, Don't Get Around Much Any More; Satin Doll. Quite simple tunes, but very enjoyable.
Other names to look into: Paul Whiteman, Bix Beiderbecke, George Olsen.
It's quite an easy-listening style, with not much improvising (not MUCH, but still a fair bit).
Swing:
Generally slightly faster and livelier than Dixieland, and focused more on brass than on woodwind instruments.
Louis Armstrong: ANYTHING by this guy. Check out the Hot Fives and Hot Sevens collections, as well as his All Stars. A wonderful singer (such a distinctive voice!) and an incredibly influential trumpet / cornet / flugel horn player.
Benny Goodman: Equally, listen to anything by this guy. He played clarinet.
Other key names: Count Basie, Duke Ellington (again), Coleman Hawkins, Django Reinhardt.
Kansas City Jazz:
We're starting to get slightly less 'easy listening' and more into an improvised music. It's still not very dissonant, but it's more experimental.
Key Names: Count Basie (again), Coleman Hawkins (again), Lester Young.
Bebop:
Now we're getting to the 'acquired taste' jazz. Bebop is fast, furious, and wildy experimental. The players thought that they'd, effectively, change the entire rules of music. And, once you start looking past the dissonance and to the musicality, it works. Also, where as the previous styles used quite large bands, bebop used trios, quartets or quintets (in general).
Charlie Parker: One of the inventors of Bebop. Tracks like Bird of Paradise, Moose the Mooche, Cherokee, and just anything he played. A genius of an alto saxophone player.
Other names: Dizzy Gillespie, Miles Davis, Cannonball Adderley, Bill Evans, Charile Christian.
Hard Bop:
An extension of Bebop, Hard Bop was ever so slightly simpler and slower (in general) but is still noise to quite a few people.
John Coltrane: Probably the pioneer of Hard Bop, his album Giant Steps defines the genre. Key tracks are Giant Steps (composed as a technical exercise but it still sounds great) and Mr P.C.
Also try and find a version of Work Song played by Nat Adderley in 1960.
That'll do for now.