Great question, one that I want constantly asked myself. At this point I am aware of two potential options (if you ignore integrations with vBulletin, phpBB, etc.) If there are others I sure would like to know.
Note the Pros and Cons are from my current understanding perspective only and certainly not a complete list:
1.) bbPress
Pros
- Blessed by Automattic thus the “official” forum for WordPress
- Similar to WordPress in theming architecture (i.e. Template Tags)
- A decent number of plugins
- Open Source
Cons
- Not a WordPress plugin but a separate app (this may be changing)
- Not very feature rich compared to many other forums.
- Clunky user-interface (again IMO)
- Frustration regarding the slow pace of improvement (nice irony here)
2.) Simple:Press
Pros
- Is a WordPress Plugin
- Appears to be actively supported
- Reasonable amount of Features
- Reasonably attractive look and feel
- Appears to also be open source
Cons
- Not being the “official” will always relegate it to 2nd class in the community
- No evidence of 3rd party plugins
- No visible financial support so if the team looses interest…
3.) BuddyPress
There is actually a 3rd option to consider which is BuddyPress. It’s more like a Facebook-in-a-box social network than a forum but might address your or other’s use-cases. It appears that it is currently getting a lot more attention than bbPress so that’s good and it’s Pros are similar to bbPress. Might be something to consider…
Anyway, those are the ones I know of any my impressions. If looking forward to any other options people have to suggest.
UDPATE:
Here’s another potential forum plugin for WordPress from Justin Tadlock. He hasn’t released it yet but I sure hope he does, sooner than later: