As I said, both MySQL and PostgreSQL are great. If you look them up, you'll see they're also both free and open-source. Each of them has its own database format, but SQL is used to create the databases and is used across the board, so you'll have no trouble porting from one to the other.
Delphi provides a number of controls you can use to make database access easier. Unfortunately, I haven't worked with them in some time and can't remember how they work or even what they're called. A quick Google found this though:
http://delphi.about.com/od/database/
http://delphi.about.com/od/mysql/
Have fun.
