That's in fact quite simple.
First, you need a web hosting. That hosting need to provide you with a decent PHP support, a MySQL base. Depending on what you want to do, where you (and your readers) are, you certainly have numerous hosting options.
In the US, you can probably try Dreamhost (other will confirm/deny it here; I have thoroughly tested only one US host and I won't recommend it). In Western Europe, OVH. If you're not sure, start with the smallest, cheapest plan you can find on a reliable, reputable, host. If need be, you will be able to upgrade in the future.
With that web hosting, you'll probably need a domain name. That's the domain.com, or .net, or whatever. Choose the one you want, and buy it from a good, reputable registrar (they sell domain name) ; if possible choose a hosting company that is also a registrar. Simpler for you. In Western Europe, OVH again.. in the US, I don't know. If you buy both at the same company, they will offer you to link one to another, without technical configuration from your part. Some host provide you with free name, like yoursite.TheHostingCompany.com or something like that. If you're unsure of your webmaster future or experience, try that, it will save a little money and you can always change it later with a real domain name. Oh another thing, check that you will be the *sole real owner* of your domain name. If you want to leave your registrar or hosting company, you'll need this (and since you pay for it, it should be yours anyway).
Overall, cheap hosting plan and a regular (like .net, .org, etc.) domain name, that should be around 20? a year. If you don't have specific needs or a lot of readers, you won't need to pay more than that (well, up to 40? maybe, but in that price range anyhow); on the other hand, choose a recommended-by people knowing their stuff, as always don't believe the random guy on the web-hosting/registrar company, even if they are slightly more expensive. The cheapest of the cheapest is usually not a good idea (even with a "professionnal" home page and pictures of yummy brunettes with headphones, you'll end up hosted in a old pentium in a wet garage, monitored a couple of hours a week by a high school boy...).
Once you pay for that, they will give you several technical data: your FTP login (used to upload files onto your web server) and password, your MySQL server, database and user name, and password. You'll need all of that.
Download from the main PunBB page the last version, and open the documentation about Install. Upload the content of the /upload directory on your webserver, by FTP. if you want your forums to be accessible at www.yourdomain.com, upload the whole thing at the root of your webserver (some hosting have a www/ or similar directory to put web files into). If you want your forum to be accessible at www.yourdomain.com/forums/, create a /forums directory in the root, and upload in that directory, and so on.
Then, just follow the documentation, it should go smoothly, if not we are here
Edit : for the technical requirement, I would strongly suggest a hosting company that provide with this (for PunBB) : PHP 4.3 or greater, MySQL 4.1 or greater, Apache, URL rewriting allowed. This will make the uses of the next version of PunBB (1.3) much easier/smoother/nicer, or even feasible at all.