A database is an organized collection of related data which can be retrieved and operated by automated means. This usually means information which you have collected on your web site and intend to use again later. An example would be a customer database which contains the customers name, address and other contact information. Another would be an orders database associated with an online shopping cart.
Simple data capture such as visitor logs and message boards rarely require the use of a database. On the other hand, almost any transaction involving money or customer data will require a database.
Databases require lots of storage and processing overhead, as well as special pages to list and maintain them. They can be a costly proposition if used incorrectly. However, used wisely, they can bring a good deal of power to a site.