Backing up your Joomla! site involves two separate processes: downloading the Joomla! files, and exporting your database.
Step 1 - Involves logging in to your web hosting account via FTP (filezilla is a popular, free program) and downloading all the Joomla! files to your desktop.
Step 2 - Log in to your hosting control panel, which is normally something like www.yourdomain.com/admin. Please note that the following screenshots are taken from the Parallels Plesk Pro control panel, but can easily be translated into other control panels, such as cpanel.
Click on the shortcuts menu followed by the MySQL Administration tool (see below).
The manage databases screen will appear. Click on the MySQL Administration Tool.
You will be prompted for your database username and password.
Logging in will launch the phpMyAdmin controls.
Select your database.
You will see a screen similar to the following with all the tables from the database listed.
Click on the export tab.
Other than the "add drop table" which needs to be turned on all default settings should remain as they are.
Finally, Select the "Save as file" option and click go.
Congratulations you have just backed up your Joomla mysql database.