Making money from a forum can be done in several ways. Here's a few ways that you might be able to earn money through your forum.
Paid Subscriptions.
If you have a popular enough forum and have something that is unique and really helpful then you can sell subscriptions to a certain area of your forum that only paid members can see.
You can also give these paid subscribers to your forum more features as well.
Paid Registration.
This is similar to the paid subscriptions but with paid registration you will be requiring your new forum members to make a small payment to join the forum.
You can make it yearly, monthly or just a one time fee. Your forum needs to have really good content though that your members would be willing to pay a fee to join your forum.
Having paid registration will also stop your forum from being spammed.
Monetize your forum with Advertisements.
If you have enough website traffic to your forum you can monetize your website forum with Advertisements such as Google Adsense, Media.net BuySellAds.com etc.
If you don't have enough website traffic then it can be hard to earn money from your forum or website with any ad network or any other ways of earning money online methods.