Shared Hosting = A server with many users sharing the server's resources VPS = A server with one then one user and limited strictly so that each have their own resources Dedicated = A server that you and only you own
I think a more clear definition of VPS might be = A server that many users share, but each has his or her own "virtual" environment that can be controlled fully as if it was a dedicated server. the advantage of VPS over shared is that you have full control of your "virtual" server - something that you don't have with a shared-host but it is less expensive than a dedicated server. However it would still suffer from performance issues from all the other shared VPS customers on the same box..
If you need hosting for a new forum. Have a shared account, as you grow go up and get better hosting. Shared --> VPS --> Dedicated.