The name of the database will be the name that you were given by your ISP for the purpose, and the name you'll have coded into your application. Databases don't really have "paths".

You will need to set up an output directory and give the full path to it for your dumps - I would suggest that you log in through ssh and use a mkdir to create a new backup directory on your server that's NOT part of you document or cgi directories (this avoids you creating a backup file that has its own URL that someone might stumble upon). The pwd command will tell you the current directory you're in so that you can work out this path.

It's also possible that mysqldump may need a full path name to be provided. If you run which mysqldump you'll be given that full path.