How do online forum sites get users?

They grow by getting connecting people that share the same interests. If they are included on a strong company/organization that alreay have lots of hits, so a forum is just one more contact channel, and grow is easier. If they don't have that advantage, they usually grow the usual way, by mouth-to-mouth/friend recommendation/ads and a lot of patience. I'm a moderator on a Portuguese programming forum and we've been growing mainly because:

1. we're very focused on what we deliver/offer and there are people interested in it - there's only one more Portuguese forum that delivers the same service;
2. we promote the development of a community - it started as a forum but know we have interesting projects, like a programming e-magazine that really attracts people to the forum and to the community, these projects got so much attention that we have requests for collaboration with printed media and lots of readers and users from Brasil.

In short: give people what they want. You may have some starting advantages if you'r a strong company/organization, but in the end it's actually the same recipe used for startups...

A few tips:

Give people something they can't get anywhere else.

Carefully structure your site architecture and HTML to optimize SEO for your target segment.

Use social media like Twitter, Facebook, StumbleUpon, etc to aggressively share good content and make it easy for your users to share content.

Get guest bloggers from your target segment to post on your blog.

See if you can link with professional organizations or other groups that might benefit from your forum.

Aggressively ask your users for feedback and suggestions on improvements and let them know when and if you'll be able to implement their ideas.

Recognize and reward your most frequent posters - they're invaluable.

Be patient - it can take a while to get to critical mass.

You'll find what may be a surprising ratio of readers (lurkers) to writers. When we launched Fastnote, we could see from Google Analytics that we were getting significantly more people who were coming just to read content.

1. Twitter (and Facebook) can help you a lot in driving people with the same interest.

2. If you want to start a community around a new forum site, I would start by looking for similar sites that are in the 'area' of what you are trying to do. You can learn a lot from watching the successful sites' tactics.

3. As for organic SEO (google will pick up your content) - I would not bet on this because it's became much harder after the recent changes Google made to their algo.

Honestly, I'm not sure they do any more - I think the popularity of standalone forums has been dropping for some time - due mostly to Facebook, Twitter, and online blog commenting. In my experience, you need at least 1000 registered users to generate a "sustainable" conversation. A low membership forum can work, however, with a well designed "alert" system, so that members get notified by email of new posts they are following.

You need posters that post quality content(can be yourself or several partners to begin with), once you have this, google will pick up on it, and everything else will come naturally.