A database is an electronic compilation of various information from one or more sources. It can be stored in almost any type of computer system, the most common Federation method being the LCARS.
In the 2260s, Starfleet vessels contained an archival audio database called voice film. (TOS: "The Conscience of the King")
In 2364, Data used the comnet database of Starfleet Command to search for hints of a conspiracy. (TNG: "Conspiracy")
The Bajoran Central Archives contain many databases, such as transit files and communications logs, which Bajoran citizens routinely access for both information and convenience. (DS9: "The Collaborator")
Romulan Senator Kimara Cretak was arrested, imprisoned, and possibly executed for attempting to access a top-secret Tal Shiar database. (DS9: "Inter Arma Enim Silent Leges")
When B'Elanna Torres was abducted by the Pralor Automated Personnel Units and instructed to build a prototype unit, she was given access to databases and schematics, but not allowed to interface with critical systems. (VOY: "Prototype")
Federation starship holodecks contained databases which stored holographic programs. (VOY: "Renaissance Man")