MediaWiki is the wiki software that powers many of the world's most well-known wikis, including Wikipedia and WikiTravel; it is also used by many wiki hosting sites, including Wikia. In our opinion, it is also the best wiki software available - free, open source, robust, reliable and internationalized.
Semantic MediaWiki is an open-source extension to MediaWiki that allows you to store semantic data in pages - database-like data that can be queried and aggregated. This is done through special markup tags; it is recommended that these tags be stored in templates.
For much more information about Semantic MediaWiki, including a full guide for both users and administrators, see the Semantic MediaWiki homepage.