There are actually 4 basic bass pickup designs.
Precision is the original fretted bass. A very simple single coil, split into 2 parts and reverse wound to try to help eliminate a little bit of the noise inherent in single coil pickups.
Jazz is a 2 pickup system. 2 Long bar magnet single coil pickups. You can blend these pickups to change the sound of the bass quite a bit. As a result, it is a more versatile system...you can get more sounds out of it.
Humbucker basses, like MusicMan Stingray Basses. My personal choice. A fat high output humbucking pickup. Less noise and more power output. A lot of thump and really deep growl are associated with them. On a musicman bass, it is accompanied by an active EQ, many other basses set up to be like the MM design use the active EQ also.
Active pickup basses. EMG bass pickups are quite nice. Several manufacturers use these active pickups, Hamer USA comes to mind. They are quiet and powerful and have a very even tone to them. They can come in Precision, Jazz, and humbucker configurations.
I used to own one Fender Jazz and one Fender Precision bass. I now own one Musicman bass. I love the sound and feel of them. in fact, there's one on Ebay that someone is going to get a steal on:
http://cgi.ebay.com/ws/eBayISAPI.dll?ViewItem&rd=1&item=7315385505&ssPageName=STRK:MEWA:IT
Not mine, but I saw it in my Ebay rounds.
H2H