The MusicBrainz Database is an SQL database that contains all the information about artists, releases, tracks, Advanced Relationships between them, but also the MusicBrainz users (editors) and the changes they entered into the database (edits). MusicBrainz currently uses PostgreSQL as its underlying database software.
The database can be downloaded from the MusicBrainz website. There are a few guides available on how to setup your own copy of the database, and import the data:
Once you've got the database running, you might want to setup a working MusicBrainz Server as well.
It is also possible to setup a mirror using MySQL: see MusicBrainz MySQL.