No you only need one domain.
You will need ro create a folder called forums under your main public_html folder. (This will explain itself once you buy your domain and look around)
You will also need a file called index.htm which will be the page that people see when they type in h t t p://www.website.com/forums
If you want link to your forum you can either set up index.htm usoing javascript to redirect them to the current location on your free website or better still move the forum to your new forums folder.
For example I have
http://www.square-bear.co.uk
and under that I have created a bootlegs directory with an index.htm page meaning I now have a
http://www.square-bear.co.uk/bootlegs webpage but all under the same one domain.
Hope that makes sense.