- Go into Cpanel and and then phpmyadmin

- you will next get the welcome screen, click on "export"

- Next you will get a "view dump" (schema) showing you the database (s)

- Choose the database you wish to transfer and then select SQL

- Hit "go", you will then get a load of text, copy all this text and paste it into a notepad file (or word) and save it on your computer as a .txt file

You now have a copy of your database on your pc, structure and all it's data.

UPLOADING THE DATABASE TO NEW HOST

- Same process, go to cpanel and then phpmyadmin

- Hit "query window" on the left, in this new window paste the entire contents of the database file (the text you copied and pasted before) into this window

- Hit "go"

You ahve now successfully transferred your entire database and it's contents, not too hard hey

For ensim, follow the same process by going to phpmyadmin

That's one way to do it, if you don't mind copy and pasting.
I don't know which, if not all, versions support making a file, but the version I use is
phpMyAdmin 2.5.3
If you want phpmyadmin to create the file for you,
- Enter phpmyadmin
- On the welcome screen, "click export"
- under structure, click add drop table
- click save as file
- click go

To load the sql file
- click SQL
- click the browse button
- find your file
- click go