I'm switching from hosting my site on Dropbox (where I switched to when .Mac web hosting stopped) to Godaddy and have a basic (very basic!) question.
Do the files for the website exist on my iMac (or connected drives) somewhere or will they only exist up on Godaddy?
Dropbox hosts files remotely but syncs them with your Mac in folders you select so it's in two places, actually. But it is not intended to be used as a web hosting platform (as docroot to a web site) nor a CDN (Content Delivery Network) although people do it for simple static sites. There are plenty of techniques to do both as you discovered re: .Mac by using it as a hosting platform. ;-)
If you must keep Dropbox for hosting the content [whatever reason] and also use godaddy's interface for domain stuff, go to "manage" in godaddy and then click on "forward domain" and paste the dropbox url. Or use a third party domain forwarding service - they use iframe and other techniques so the world would see your domain URL and never know. I am not sure if godaddy's domain forwarding hides the dropbox URL or not, hence this suggestion.
On the other hand, if you want to switch entirely to godaddy then the response from maflynn is what you need to do, and do so from your local mac folders.
Using just godaddy as your do-all webhost and domain manager is probably the best long term solution in terms of security, simplicity, scalability and performance although not free of course like dropbox.