Databases are used to store information, especially information that is frequently updated. It is much more efficient than storing information in a static HTML or text document. Most modern websites use databases in one way or another. Popular content management systems like WordPress, Drupal and Joomla are all database driven. So are shopping carts and forum. This website is even database driven.
When installing a script or web application, a database often needs to be setup. Depending on the script, you may have to create or setup the database during the installation process.
Most web hosts use MySQL, which is the world's most popular open source database. MySQL is a leading standard in today's web. Download whitepapers and explore at http://www.mysql.com