phpBB - is probably the most widely used forum script available today. Simple to use while still giveing you great features like other paid forum scripts have.
You can also be sure your users will feel right at home with this script since it's so widely used.

phpBB resources - free themes, tutorials, KB & user guides, forums

SMF - is said to be the most secure free forum script and also has some very interesting features.
For slightly more advanced users that have more advanced needs.

This forum script has a little more of a learning curve than phpBB but does have a few unique features that some forum admins will appreciate.

SMF resources - themes, mods, support

vBulletin - is the most preffered and advanced paid forum script available.
This software is not for novices. Only people who want complete control and incredibley advanced features should try this software.
Although the admin side is advanced the user side is very simple and makes it easy for everyone to make full use of all the great features.

vBulletin resources - purchase vBulletin, support, forums

Invision Power Board - has inspired quite a loyal following. Some would even say that they prefer it over vbulletin.
For slightly more advanced users that have more advanced needs.
You may have a tough time deciding between IPB and vBulletin. They are both very good forum scripts that have alot of advanced features.

IPB resources -purchase IPB