A database is a collection system for storing and retrieving large amounts of data. Many content managed websites are built on databases because they allow large amounts of data to be managed in different ways.
Databases are organised into a series of tables. Typically you will have a 'customer' table which holds all of the basic information about your customers. Then you will have an 'orders' table which details all of the orders they have made with you and a 'products' table, listing all of the products you stock. These are some of the most common basic tables used in database design. Databases often hold many other tables for different purposes.
When accessing data through a website, the data is accessed using a dynamic lanugage such as PHP or ASP.NET.