How do I get past, "An error occured while backing up the database:, , Could not create *"?

You might receive this error if you have customize the backup path to a location that TntMPD is not allowed to write out to.  To revert to the default backup location, under the Tools menu, click Options, then click the Installation tab.  Change the Backup option to "To This Computer":

For an external backup solution, it is strongly recommended that you put your database on Dropbox:
http://www.tntware.com/tntmpd/faqs/en/how-can-i-share-my-database-with-my-spouse-over-dropbox.aspx