Hi,
To answer your question, I would look at it in reverse - Firstly I think the most useful websites (or desktop applications ) are 'data-driven'. This means the web site has a basic frame work of pages - generic in design - but these are populated by the users interaction.
For example, I could log into your web site and the database would hold information about who I am, and what I am entitled to see - my profile. It would then populate the generic pages accordingly; May be it would only show me the news on your site. When you log in using the same pages, the information in the database lets the web site know that you are an administrator. So instead of just seeing the news, you can edit it.
This is a very basic example, but I think demonstrates it quite well.
Regarding learning it, you must consider SQL is a language that does have a standard, but taking four of the most popular databases around - SQL Server, Oracle, Access and MySQL you will find syntactical differences between them. They also have their own strengths and weaknesses.
For a small website I would look at Access. It is quite straight forward, but on the downside you would need Access (the office program) or Visual Studio to use it. It also has a problem with concurrency - the number of simultaneous users. I have heard conflicting stories but I believe the truth lies somewhere between the 6 and 20 I have been told its capable of.
Both Microsoft SQL Server and Oracle have free express versions. These scale a little better than Access, but are a little more sophisticated in nature. That said, you can get Visual basic.NET express or Visual Web Developer Express as free downloads from Microsoft, and they can also be bought with some reasonable books to get you started (Wrox and Dummies) Both allow you to use the databases - Access and SQL Server - from within them.
I cannot comment on MySQL as it the only one I haven't tried. As a professional I would advise you looking at the Microsoft SQL Server option because its well supported, and has plenty of free stuff and good tutorials. Oracle is just as good, but has been aimed at the large scale databases.
Anyway - hope this all helps.