Are forums indexed by search engines?

Yes. Most forums are public and can be indexed by search engines. In general, it is recommended that anyone using the Internet assume that anything they post online will be viewable by all people for the rest of time. 

One exception are the private forums, which may be requested by members, and which can only be accessed by individuals who have been granted access expressly by the member(s) who own the forum. Go here for more information on how private forums can be created.