Here are my favorites.
vBulletin (vB) is one of the most popular premium forum software on the web, and the choice of many big websites. Some of it's built-in features include photo albums, groups, calendar, events management, polls, blog and more.
phpBB is the excellent free and open source forum bulletin board software. Its contains hundreds of style and image packages to customize your board.
XenForo is a popular paid forum software package written in the PHP programming language and developed by former vBulletin lead developers. It has built-in SEO features, powerful add-ons, recent activity stream, social engagement functionalities, etc.