Jazz is a very broad category of music. What characterizes jazz music is the emphasis on chord structures (which are often complex) and improvisational soloing. Having said that, the many styles of jazz lend themselves to a very diverse listening experience.
The best thing to do would be to get on iTunes, go to the iTunes store and click on iTunes essentials (listed under More in Music from the home page). Then go to iTunes Essentials - Genres and History. There are many different iTunes essentials collections for various jazz styles. These collections include:
Brazilian (these feature African/Latin American rhythms)
Contemporary Jazz (These are often the most complex and abstract, with much emphasis on improvisation. They also incorporate electronica)
Discover Jazz: Big Bands (This more traditional style featured large jazz bands, with more structure and less improv. Many of these were dance tunes [swing])
Discover Jazz: Standards (These are the most famous jazz songs of all time)
Dixieland Jazz (Dixieland jazz is very traditional-sounding, yet fun and spontaneous.)
Future Jazz (some of this borders on atonal, electronic music, you may not like this)
Gypsy Jazz (Jazz with that exotic Eastern influence, leading to very interesting melodies and rhythms)
Jazz 101 (Basic jazz songs)
Jazz Fusion (Jazz blended with rock and electronic elements)
Jazz Guitar (jazz style guitar is nothing like its rock counterpart; it is more mellow and melodic)
Jazz History 1: Big Band & Swing (similar to the earlier big band collection)
Jazz History 2: Bebop and beyond (Bebop came after swing. Performers started to explore more complex melodies and chords)
Jazz History 3: The Modern Era (When Jazz became even more complex and varied, but sometimes simple)
Jazz Saxophone (speaks for itself)
Jazz Scat Vocals (You should be aware that a large part of jazz is improvisation, or the making up of melodies during the performance. Scat singing is when a singer is improvising a melody and vowels/consonants to form nonsense words. A very interesting style)
Jazz Trumpet
Jazz Vocals (this would be more standard singing, but in jazz style and backed up by jazz ensembles)
Jazz: The Young Lions (this is an 80's jazz movement to get back to the basics, I think)
Latin Jazz (similar to Brazilian in many ways, but different as this is the Spanish-speaking Latin America's jazz)
Mystic Jazz (deep stuff)
New Orleans Jazz (similar to Dixieland)
Piano Jazz (the piano, with the ability to play chords and melodies at once, is one of the most indispensable jazz instruments. Piano solos, which involve chords, not just notes, are among the most interesting)
Rising Stars: Jazz (popular jazz musicians today)
Women of Jazz (female jazz singers have that characteristic rich, warm, husky voice)
In each collection, there are three sections:
The Basics, to see if you like the style
Next Steps
Deep Cuts
The smallest collections have over 20 songs, and the largest well over 50. All the songs are purchased individually. You will start to find the styles and the artists you like.
Here are a couple of my personal favorites:
Swing/Bebop style:
Sing Sang Sung/Gordon Goodwin's Big Phat Band/Swingin' for the Fences
Anything by Milt Jackson (he is an amazing jazz vibraphone player)
Jazz Fusion:
Birdland/Quincy Jones/Back On the Block (most of the rest of this album is not jazz, however)
Brazilian:
anything by Antonio Carlos Jobim
Singing Jazz Groups:
anything by Manhattan Transfer
anything by New York Voices
Modern Jazz:
Go Fly a Kite/Dave Holland Quintet/Not for Nothin'
Cool Jazz:
Giant Steps/John Coltrane/Giant Steps
So What/Miles Davis/Kind of Blue
Just a very few specific suggestions. But iTunes essentials should be a lot more helpful!
Oh, and jazz arose from blues, but it is much more complex. Blues has simpler chords, and blues improvisation is based on the melody. Most jazz improvisation is based on the chord structure.