Wikibase is the software that enables MediaWiki to store structured data or access data that is stored in a structured data repository. Wikibase basically consists of two MediaWiki extensions, Wikibase Repository and Wikibase Client, that can be enabled individually or together for a certain MediaWiki installation to turn it into a structured data repository, a client of a structured data repository or both.
For example, Wikidata is a Wikibase Repository as well as a Wikibase Client.
You can find out more about the overall architecture of Wikibase and its components on http://wikiba.se.
Wikibase Repository is a MediaWiki extension that can turn a MediaWiki installation into a structured data repository. It allows editing and storing of that data.
Wikibase Client is a MediaWiki extension that can turn a MediaWiki installation into a client of a structured data repository. It allows to use and display data from a Wikibase Repository via Lua modules or parser functions. Clients can also use centralized language links or article placeholders.
In theory, Wikibase can be installed just like any other MediaWiki extension. However, due to complexity and dependencies, it requires some additional steps. Please refer to this installation guide if you want to setup your own Wikibase installation.