MediaWiki
MediaWiki is the software that most wikis (including All The Tropes) run on, and it is a play on the name "Wikimedia" (as in the Wikimedia Foundation).
In fact, MediaWiki was designed by Wikimedia Foundation to replace UseModWiki, what Wikipedia initially used until it's limitations quickly became apparent, so they created a new wiki engine that was written in the PHP scripting language with a MySQL database backend, designed for hosting a community based encyclopedia project (or any large scale wiki project in general).
MediaWiki is licensed under the GPLv2 license and can run on multiple platforms. It is free software that can be modified by anyone for any purpose, and it generally tends towards open source data formats, and its licensing terms were chosen to reflect this.
The most well known MediaWiki wiki would be the Wikimedia Foundation's flagship wiki, Wikipedia, from which other projects such as Wikitionary, Wikinews, and Wikibooks were designed as spinoffs. MediaWiki is also used by many companies for writing help documentation, used by Wikia (in a heavily modified format) to host wikis of all varieties, and can be used by practically anyone to host a wiki for any purpose one can dream up.
- Wikipedia Syntaxer - Technically incorrect, as Wikipedia uses the MediaWiki markup language and thus this should be called "MediaWiki Syntaxer", but Wikipedia is what this trope is most affliated with.