When the web was young, web pages were static - that is, they didn't change. Now we use databases and DMTML to make pages dynamic, capable of reconfiguring themselves based on the user's preferences, capable of presenting real-time information, and using the power of computers to do what computers are good at.
We use databases to provide current info such as weather or traffic on the website. When the database changes, the website changes.
Databases are also used to administer eCRM (Customer Relationship Management) programs, such as frequent-skier programs and bulk emailing campaigns.
The primary tool we use for database-backed websites is ColdFusion, which connects traditional SQL Access databases with web servers.