In phpMyAdmin the create database form is found on the start page. Creating the actual database is simply a case of choosing the database name and pressing create. This obviously requires the user to have CREATE privileges.

To backup a copy of your database, choose the “export” menu item.

In the support tab you can choose which tables you wish to backup (presumably all off them), and if you want the table structure or data, or both. Obviously the data is the important part, but including the structure as well makes restoring a backup much easier.

You can then choose to export the backup either uncompressed, zipped, gzipped or bzipped.