GrimForum is a good-looking forum system with professionally designed graphics. Colours are user-definable, and there are many different admin levels for members. It runs fast, with lockable topics private messaging, and more.
SowiBB is a bulletin board application written in Object Oriented PHP, with MySQL backend. If you want a fast, easy to use, lightweight bulletin board, you've found the right one. SowiBB's installer requires no paths what so ever, and just requires some names and info to complete the installation. Also, you can download a tour if you want your members to be able to find out how to do stuff.
ADP Forum is a discussion board system that doesn't use any database, but instead stores all the data in txt files. The forum is highly customizable, you can easily edit: colours, fonts, icons. There is a detailed readme file that explains how to install. It features: user authentication, member ratings, avatars, configuration panel, administration, email notifications, smiley option, edit and delete options. Languages available: English, Italian, French, Spanish, Swedish, Polish, Finnish, Norwegian, Slovakian, Bosnian.
PHP/MySQL-based board that runs very stable extremly fast even on large sites like www.thewall.de/forum/. Features: MySQL Support, usergroups, private messaging, news backend with comment function, avatars, statistics, multilingual installer, highly customizable with themes and hacks.
An easy to use, and easy to install forum application. Visitors don't have to register, but can enter an username and password to edit their posts. You can configure forums to be validated, meaning each new topic (not new reply's) has to be validated by the webmaster/admin.
ProZone is a fast, feature-packed message board system that is easy to install, use and moderate. One master admin can moderate all threads and change settings. The Master Admin can make unlimited extra admins for single or more threads.