Many web sites have connections to databases. Sometimes this is obvious, because one feature of the site is the ability to search a database. Sometimes that database features are less obvious, perhaps because the database is actually powering some form of content management system.
Sometimes databases or other programs play a part in generating the content which appears on a web site. Its also possible to have database-like functions on a web site which are in fact provided either by the search functionality of a web browser, or by google.
Its generally worth adding a full database to a web site only if that is really necessary. But when it is the right thing to do, we have handled a variety of database projects using PostgreSQL and MySQL, for online searches, content management and data processing.