A database is basically a centralized repository (or simply a store house) where put store all the data which is required/handled by your organization. It may be directly linked to a single application (like for example a website which sells products) or there may be multiple internal applications sharing it. So any info you capture, any info you want to set and display to your customers etc are all the data which gets stored inside the database.
The way it works is by means of a programming language called SQL (assuming traditional RDBMS). Applications connect to your databases using connectors and execute SQL statements which will either manipulate (store, modify, delete) data or select data from your database