1. Open a new Terminal Window.
  2. Type in the following command: mysqldump -h localhost -u root -pPASSWORD database > backup.sql
  3. This will backup the database to the file backup.sql
  4. -h = Hostname -u = Username -p = Password, no space between the p and the password allowed

NOTE: Has been tested on Ubuntu 9.04 and mySql 5.0.75