Mysql databases can be backed up by doing the following:

Go to 'Backup', in cPanel. 

Underneath where it says "Download a SQL DB Backup", click on the database name that you wish to backup. 

Select a place and filename for you to save on to your local computer 
Do NOT attempt to open this file with Winzip.

You have now successfully backed up your Mysql database with Cpanel. There are other ways to accomplish this task, such as doing a dump from PhpMyadmin, or doing a dump from the command line, but the cpanel method is quick and easy.