Monetizing a forum can be tricky, brakkar. Advertising is certainly one way, and that can include:
1) Regular CPM or CPC ads.
2) Google Adsense.
3) Affiliate ads/links related to your topic.
Adsense is the easiest to get going, but the others may be more lucrative if you are in a good niche.
Selling premium memberships is fairly common. Some of the benefits you see often are,
- private forums
- more storage for PMs
- recognition (e.g., bigger/better avatar, different user class, etc.)
- more board powers, e.g., can post attachments, pictures, etc.
I'd guess the exact benefits that will appeal most will vary greatly by site - a gamers board is going to be quite different than an executive recruiters forum, for example.
What kind of off-site stuff did you have in mind?