MySQL (pronounced "My ess cue el") is an open source Relational Database Management System that uses Structured Query Language. Information is stored in "Tables" which can be thought of as the equivalent of Excel spreadsheets. A single MySQL database can contain many tables at once and store thousands of individual records. It's fast, reliable and flexible.
PHP and MySQL work extremely well together. Useful tutorials on how to use them are freely available on the web and some excellent reference books have been produced catering for complete beginners to experienced web application developers.