Well, From the top of my head (meaning prolly wildly inaccurate) a database is simply a store of data, that data is stored in tables each table has many rows each with different names witch take different values INT, TEXT perhaps.
On the internet these are used alot, say you have a online shop, the details on your stock could be stored in a table in a database with the row names being StockName, ItemNo (this would be a primary key (don't worry about that)), Price, etc...

Then using a serverside language like PHP or ASP.NET you could format a page that listed the contents of the database in an intuative (sp) way.

So, a database is a useful way to store data, its faster and often easyer then using plain text files.

SQL is the Structured Query Language, alot of databases use this as a way of well interfacing with them, SQL is quite similar across different databases (there are some slight differences) so learning SQL really allows you to interface with alot of different databases. The tutorial shows you some of the things you can do with SQL.

I hope this helps you out