SQL is an acronym for structured query language and it is a language designed specifically for accessing databases to read or write data. SQL databases are the most popular type since it is incredibly versatile and easy to use. Most people use them for websites, although they are also used for non-web applications.
Several popular SQL packages exist, but the most popular is mySQL, which is open source. Microsoft also provides SQL server software, but it is rather expensive. For security and cost reasons, mySQL is probably the best choice if you plan to use SQL for web design.
With SQL, you can create as many databases as you like. Inside of a database, there are objects called tables. Inside of tables, there are objects called rows that have columns containing different types of information. For example, you could design a table so that each row has a column with an integer, some text, and a category. Designing a table is like designing a class in C++ or Java, since you are essentially creating variables that can be different for each individual instance of an object.
The variety of SQL commands gives the software a lot of capabilities. You can, for example, query a database and have it only return rows in a table that have a certain value for their category. This way, you can easily access information of varying types within a single table. It also allows you to create dynamic pages using PHP. Instead of hard-coding links to every single object in a table, like you would have to do in HTML, you can write code the queries the table for certain values and automatically creates links. In the end, this can save the website designer thousands of hours of work.