When it comes to backing up a joomla website's MySQL database the tool most used is phpmyadmin.
To use this through any website, it can be accessed by going to your webistes cpanel.
This is usually located on www.yoursite.com/cpanel
Once you have logged in scroll down to the database section and click on the phpmyadmin icon.
Once you have clicked on the icon select the database from the drop down list which you use for your joomla website.
If you have forgotten the name of the database you can find this out by going to the global configuration settings of your joomla website, click on the server tab and you can find the name of the database used.
The next step once your have selected the database from the drop down menu is to click the export tab.
By default it will select all of the tables in the database to export, if you wish to export a single database and not the whole database you can select items by holding down the "ctrl" button and selecting the items you wish to export.
The next step is to choose how you wish to save the file and give the database a name. Unzipped, zipped or gzipped are the options you have to choose for the exporting and saving of your joomla database.
Once you have named the file click the go button and a save box will pop up. Save the file and you have now exported your joomla database!