To back up your MySQL database, we're going to be using phpMyAdmin, a web software that manages MySQL databases. Let's begin.
Go to the URL: http://mysql.naughtybananas.com on your web browser. You will be prompted to enter in a username and password. By default, this is set to be your FTP username and password.

After you have logged into phpMyAdmin, click on the link 'Export'. This will bring you to a page, that allows you to export (back up) your MySQL databases.

There is a list of databases on the left side of the page. Select by clicking on the database that you wish to export. To export multiple (or all) databases, hold onto your Ctrl key when selecting individual databases.

Ensure that the radio box for 'SQL' is selected, as shown in the image above.
Now scroll down to the bottom of the page, you will see an option to save the database as a file. Check on 'Save as file', and enter a file name for 'File name template'. You may change the '__SERVER__' to a file name that you wish.

Select all other settings as per the image above. All other settings can be left as they are. Click on the 'Go' button once you're ready, and you should be downloading your database back up already!