ForumMatrix - Compare forum software side-by-side
If you aren't sure where to start ForumMatrix has a handy Choice Wizard that will ask you a few specific questions about what features you may be looking for. I can see this as especially helpful for people new to running a forum. It filters results down to flat versus threaded, license type/cost, storage type (MySQL, PostgreSQL, Microsoft SQL, etc.), attachments, and more. If you still aren't sure and need advice from other users, you can also drop by their own forums and consult others on their own opinions.