SQLite is a free and open source SQL relational database management system created by D. Richard Hipp and first released on 2000-08-17. It implements the bulk of the SQL standard in a cross-platform C library which can be stored in a single file and accessed without needing a client/server setup, so it's perfect for data storage for standalone programs and has been used by thousands of developers world wide, is included in hundreds of programs, and has bindings for scores of computer languages.
If you plan on using SQLite, you should also check out the powerful SQLite viewer program, DB Browser For SQLite.
I first used SQLite while looking for a database backend for a C# program I was writing. I was already familiar with a couple other SQL implementations like Visual FoxPro and MySQL, but I wanted to try something that was entirely free and fully standalone. The database ended up being prefect for my program.