Trying to define jazz is going to be a bad time, like trying to define art or the meaning of life. Jazz originated and developed over time as an African American take on popular music; bands playing popular tunes all night at dance clubs developed their own style, swinging the rhythm to keep people dancing and improvising to keep the music fresh.

I think the development of bebop had a huge effect on jazz, taking it from being primarily dance music to then focusing on virtuosic players. Bands got smaller and soloists became much more of a focus. Since bebop popped up in the 40s, most jazz has been heavily influenced by it.

Since the early 60s probably, there's been a lot more branching and specialization in different jazz traditions, along with the growth of free jazz and other avant garde styles that explore breaking various "rules" of trad jazz.

It's hard to recommend more than a few certified classics without knowing what you're looking for. I'd say early John Coltrane, Hank Mobley, and Charlie Parker are a great place to start. If you want to go earlier, Louis Armstrong's early stuff is great if you can get over the recording quality. If you want to go later, good luck, there's a ton.