The initial post is correct in that this is a very broad question.
It really depends on the jazz you're listening to! Summing up jazz into this singular box of "jazz" really doesn't give the genre the credit it deserves, as there are forms of jazz to which the only common thread is instrumentation.
As a broad rule of thumb, jazz has a lot to do with improvising over a given form, and expanding upon the given melody and chord progression to create something new and explore the space you've been given.
An important point to note, is that jazz theory (and all theory for that matter) are not rules, but guidelines, and the best attempt at describing what musicians are doing when they play; A lot of Jazz musicians really aren't thinking about theory when they play, but thinking about the music itself.
I would recommend checking out Miles Davis' "Nefertiti" (the album), Wayne Shorter's "Speak No Evil" (Also an album), and Art Blakey's "Free For All" (once again, an album).
"A creative artist works on his next composition because he was not satisfied with his previous one." - Dimitri Shostakovich
Like people have been saying here, there are too many things that jazz jazz to be able to pin it down concisely, but I think one of the main characteristic things about jazz chord progressions is that they will often spend short periods of time in keys outside of the home key. e.g. The first bar of a tune might be in C, but then the 2nd and third bars will be in D minor, and then in the 4th bar it moves back to C again.
Mark Levine's Jazz Piano Book was mentioned here before, and it is a great resource. Mark Levine also has a book called The Jazz Theory Book, which is also great, and a little more general. He really lays out jazz theory in a very plain-spoken and easy to understand way in both of those books. I recommend them.
@[Only registered and activated users can see links. Click here to register] I don't really know of sites that are all around jazz music but I can give you artist to listen to.
Also put these artist into Spotify and start a radio for them and you'll get similar artist.
I recommend first learning the following scales, using "imaginative memory" to lock them in your mind: Ionian, Dorian,Phrygian, Lydian,Mixolydian, Aeolian, Locrian.
There is an iPhone app that is pretty buggy and old, but is a great start as a tool to help with getting these in your mind and their sound or flavor as well. I was hoping for an update at some point. Since I can't post links, look for the app called Guitar and Ear, released in 2012.
Then, to see how these scales link up and build on each other, Pat Martino is your man. The way he uses "geometry" to link the scales is seriously genius. This guy will blow your mind. I can't post links, but try Linear Expression by Pat Martino.
There's an italian jazz piano player turned comedian who said: "Jazz is when you're playing a beautiful melody and then start to hit random notes that do not have anything to do with it."
Heh, not entirely wrong, with some schools of jazz. I had a jazz guitar teacher who would complain about what he called "jazz nazis", a group of people who "engaged in contests to see who could use the most hideous and unforgivable chord in a song" and went to concerts so they could be amazed by the absolutely dreadful chords their fellow jazz nazis were playing. A typical (exaggerated) jazz nazi progression is something like:
That said, modes and modal playing are a good start, as well as getting a handle on what sort of mood each evokes, as well as the moods evoked by changing from one to another. That said, speaking as a guy who knows a lot of music theory: in the end, music theory is not math, it's like color theory-it tells you what might look good, but it doesn't describe everything that looks good, and in the end it's more like a set of guidelines - in the end it's really more an ad hoc description of what sorts of playing tend to sound good, and while it's helpful, it is not a substitute for playing a lot and figuring out what sounds good to you.
C 7 = C, E, G, Bb
C 9 = C, E, G, Bb, D
C 11 = C, E, G, Bb, D, F#
C 13 = C, E, G, Bb, D, F#, A
However. . . note that you can have any of the above which omit one or more of the inner intervals, and more importantly, the voicing you choose will make them sound more or less "jazzy." You may need to mess around with them to work that out.
(edit: nb. # 11 is the most common voicing in a dominant chord (ie flat 7) so that's why I put it here, but you can make dominant and also minor chords using natural 11. For a major chord (not dominant), the 11 is normally sharp.)