[gulldarek]

If you've cpanel then ssh too. Assuming that you're MS Windows user, download PuTTY from here: http://the.earth.li/~sgtatham/putty/.../x86/putty.exe . Run it, enter your servers ip and port, click "Open". You'll be asked for login and password (login as root if you can). Then you've to change dir to your public_html (probably /home/_login_/public_html/. _login_ is your account name. From there run this command:

mysqldump --opt -Q -u dbusername -p databasename > backup.sql

dbusername is a login to your database and databasename is a name od the db ;-). you'll be asked for password, enter it and wait until you see a cursor which means that the data from db was dumped to specified file (backup.sql). Now, point your browser to http://www.yourwebsite.com/backup.sql and download it OR if you've shell access on your new server connect to it and use wget to download dumped data. Tell me if you need some live help, I can help you.