The most widely used free database is MySQL. It is both efficient and powerful, and has hundreds of thousands of fans all over the world. Most common scripting languages can be used with it, but the combination PHP/MySQL is most popular. Because of this, PHP/MySQL-hosting is most common on Linux servers, and since Linux, MySQL and PHP are all free and very efficient (less strain on the server), PHP/MySQL hosting is cheap.
There are plenty of MySQL-driven PHP scripts available for free on the internet, like discussion forums, directory scripts and shopping carts. MySQL can also be installed on Windows as can PHP, and these are often used in a development environment so you can develop on your local machine without an Internet connection.
