First, stop the MySQL server, this is dependant on how your system operates. For a Debian based box (Ubuntu, Knoppix et al):

sudo /etc/init.d/mysql stop

Next, start up MySQL and tell it to bypass the authentication tables (the & is to background the process):

mysqld_safe --skip-grant-tables &

Launch the MySQL client:

mysql -u root mysql

and finally change the root password:

update user set Password=PASSWORD('new-password') where user='root';
flush privileges;
exit;

Kill the running MySQL server and start it again as per usual.

Content copied from http://www.howtoforge.com/reset-forgotten-mysql-root-password