I have a forum which runs on phpBB3, but I believe that if you don't mind paying a bit, vBulletin is awesome.
You would still need to pay for hosting with either though. There are some free, but to be honest, I haven't had very good experiences with any of them.
I use Simple Machines Forums because I couldn't stand the amount of spam I was getting on my phpBB. I'd recommend it and it was actually very easy to set up manually.