I run my own web design company and we use PHP/MySQL with Linux. It's great because its open source (free) and is extremely powerful and quick to develop.
Basically databases are used to store large amounts data that can often be divided into tables and fields. They are used for mailing lists, business directories, ecommerce, and basically anything else where you would need to be able to manage huge amounts of data.
The disadvantages are that it can be complex and time-consuming to setup initially and because of this it is often very expensive. Even the database programs themselves can be very expensive for commercial solutions (i.e. Oracle, DB2, Sybase). These can all cost over a million dollars for high end systems.
Many Internet hosts allow some form of database, usually MySQL on linux or Microsoft SQL on NT. Of course, this doesn't include developing the application which is where the bulk of the time/cost is.
If you are good with HTML, javascript, or any other programming, it is not a huge stretch to learn how to develop these databases. There are many good tutorials, but some of the best I have found are:
An Introduction to MySQL
AND
Building a Database-Driven Web Site Using PHP and MySQL
These will definitely get you started in the right direction.