A database, in general terms, is a bank of information stored electronically. In Football Manger terms, the database is where all the information about almost everything in the game is stored, from all the players' profiles and skills right down to the minute little details like how windy it is in South Africa in the winter.
Football Manager's database can be explored and edited using the official data editor or with other third-party utilities available from various websites. Although editing the game's database is not very difficult, it is advised that you only attempt it if you are familiar with the game and you should always make a back-up copy of the original database in case something goes wrong during editing.