MySQL is a popular database, with many features and is available on most servers.
Jig is a lightweight and fast database engine that uses flat files as storage containers. This makes it super fast, since you don't need wait for the DB service and your server can keep the whole DB files in memory.
This is a good choice for starting, but could become slow when the DB files grow (1000+ records).
SQLite is a very fast and small SQL database.
PostgreSQL is a feature-rich SQL database that runs on many operating-systems.
SQL Server from Microsoft (2008 or newer)
MongoDB is a fast, object-oriented NoSQL database.