It will all depend on what you have available server-side and where your skills lie. There are classic forum scripts that you have to pay for like UBB and vBulletin but the 'clones' available are just as powerful. Anyway have a look through these:
External
Technically these don't require any hosting or scrpting knowledge but fit in well here:
ezboard
PHP
phpBB
InvisionBoard - Super Fast, Well Coded, and Free!
Phorum - Simple, Free.
hotscripts.com/PHP/Scripts_and_Programs/Discussion_Boards/
General
Google directory
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Relevant threads:
Alternatives to phpBB - redux
Forum software (emphasis on simple, standards-compliant forums)
UBB forum
UBB-style forum software
I'm looking for a free Discussion Board scripts in PHP
a good free cgi forum system
Favorite PHP Message Board
need a good discussion board script in PHP
Bulletin Board Software
Forums
best free php forum?
Message Board
bulletin boards
Php Message Boards: Recommendations and Beginner Questions
free PHP forum
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Relevant FAQs:
Can you give me help installing/modifying phpBB?
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Emperor
(Added by: Emperor on Fri 13-Sep-2002)
(Edited by: Emperor on Fri 13-Sep-2002)
(Edited by: Synthetic on Mon 16-Sep-2002)
(Edited by: Rick on Tue 01-Apr-2003)
(Edited by: Emperor on Fri 23-May-2003)
(Edited by Slime on 08-20-2005 18:43)
(Edited by Emperor on 01-14-2006 16:16)