MySQL is the hugely popular SQL database server. It is the only part of MAMP that isn't installed by default with Leopard. No problem, a quick visit to mysql.com will let you download it, select Mac OS X package format. (N.B. Pick the 32-bit version, the 64-bit will cause a headache later if you plan to use Perl's MySQL DBI, see my previous post).Run all 3 applications within the download: mysql-5.1.X-osx10.5-x86.pkg, MySQLStartupItem.pkg and MySQL.prefPane.
MySQL can be started/stopped via the the MySQL panel in System preferences, or via the command-line:
sudo /Library/StartupItems/MySQLCOM/MySQLCOM {start|stop|restart}
Now it is installed you should remove the anonymous login accounts and add a password to the root account. There is plenty of information on securing MySQL available here.
Configure PHP in part III.