IMO you'll never be able to define what jazz is - it's a constantly evolving/growing/changing art form - and at one point in time, some may consider something jazz while others will not. As bigshot mentioned, improv is normally one of the universal defining characteristics of jazz. Some will argue that is must also swing, but there are just as many, if not more, who will argue that a swing feeling isn't necessary. Jazz is always reaching "outside the box" so trying to define it, or put it in a box, is nearly impossible. I think it was Eric Dolphy who once said that jazz is like an octopus - it has all these arms and just reaches out and grabs what it wants; meaning that jazz artists will grab stuff from hip-hop, classical, rock, blues, and any other genere, and incorporate it into jazz.
That being said, you have tons of options and if you could talk about your preferences, I'm sure a lot of people will have suggestions (for example, do you like vocals with your jazz? want big band/swing recommendations? what about favorite instruments?). Personally, I think the straightahead jazz style is a great starting point because it's what most people think of when they hear the word jazz (they either think that or avant garde jazz). Straightahead jazz usually has a 4/4 time signature and walking bass which give it a swing feeling. Below are a few recommendations I usually give to friends who are just getting into jazz.
Christian McBride & Inside Straight - "Kind of Brown"
Donald Harrison - "Indian Blues"
Joshua Redman - "Spirit of the Movement: Live at The Village Vanguard"
The Bill Charlap Trio - "Live at The Village Vanguard"
New Orleans style:
Rebirth Brass Band - "Hot Venom"
Trombone Shorty - "For True"
Below is a link to my website for jazz reviews - I put a lot of emphasis on writing reviews that new/novice jazz fans will hopefully find useful; in other words, I try to avoid a lot of technical jargon and concentrate on the music as I hear it. Whenever possible, I try and make comparisions to other known artists (both jazz and non-jazz artists) and I usually discuss who would enjoy a certain CD - meaning, is it something for everyone or more suited for the "seasoned" jazz fan. Finally, I focus primarily on the contemporary jazz scence as there is truly a huge pool of talent making great music today!