MySQL is one of the most popular relational database systems which is widely used with PHP applications. It's relatively easy to set up and use. Here's a quick guide of how to install and configure MySQL in the newest release of Ubuntu.
First install the MySQL server:
sudo aptitude install mysql-server
The installer will ask you for a root password. Using root user, however, is not the best practice. It's way better to give yourself permissions to create and use databases.
First open MySQL console:
mysql -p -u root
Next, grant yourself all priviledges for localhost connections (replace 'kuba' with your user name and password with password of your preference):
GRANT ALL PRIVILEGES ON *.* TO kuba@'localhost' IDENTIFIED BY 'my$ecret'; \q
Now you can create MySQL databases without the need of using root account:
mysqladmin -p create wordpress
Note: Read about PostgreSQL in my another post: PostgreSQL installation in Ubuntu 10.04.