MediaWiki is the name for the Wiki engine that runs the Wikipedia and other Wikimedia projects!
Wikipedia was previously using UseMod as its WikiEngine, hence MediaWiki is largely derived from UseMod, with specific enhancements focused on scalability, like a SQL backend, output caching and section editing. Its MediaWikiText markup is very similiar to UseMod's markup.
Its project homepage is at http://mediawiki.org/ and it is discussed also at http://meta.wikimedia.org/wiki/MediaWiki.
For similarities/differences with MoinMoin, see MoinMoinVsMediaWiki and WikiEngineComparison, MediaWiki feature list and MediaWiki reviews.
"MediaWiki is primarily targeted at Wikipedia and Wikimedia's other projects (and other similar large-scale sites with people running their own servers), secondarily at people running local instances to work with data from our sites, and only incidentally at anyone else." -- Brion Vibber (MediaWiki developer)
MediaWiki plugin
There is a MediaWiki plugin (parser) for MoinMoin named "media4moin" that makes Moin compatible. You can download it at ParserMarket. It enables you to use most of MediaWiki`s markup, but some (like InterWiki support) is missing.
Other resources for moving from one engine to the other are listed on the mediawiki.org page on MoinMoin.
see also MediaWikiConverter, WikiPediaForMoinMoin.