Having your website mentioned in forums is a powerful way to promote it in front of people with needs related to the content of that website. The forum promotion is totally free! The forum is a place where various people may sign up (for free) and open topics for discussion or reply to already open threads. You have 2 basic ways to get exposure in forums through:
Your Signature
Upon sign-up, the member usually is requested to choose a nickname and a signature. It is really smart to use that opportunity and have your website's URL included in your signature. You may even use your nickname and associate it with your domain, or the topic of your website.
Your Posts
If you manage to write interesting things and write as often as possible, you become one of the highly reputable members. Thanks to the quality and quantity of your posts, you will attract attention to your website as well.
Note that advertising is not allowed in most forums. Unless you want to be banned, or the forum rules allow it, you better not try openly advertise your website.