The following is an explanation given by Sean the software developer.
With "Dropbox Connection" Moneydance uses the Dropbox API to talk to the Dropbox service directly. So, you don't need to have Dropbox client actually installed on your computer, but you do need to be online when Moneydance is running in order for changes to be synced. With "Drobox Folder" Moneydance will just read and write files to your Dropbox folder on your computer, which means that you need to have Dropbox client installed and running, but you don't need to be online in order for Moneydance to read and write the changes.
I believe Shared Folder depends on you having a folder on your computer that you are already sharing with another method other than Dropbox.
Probably the best method is Dropbox Folder which requires you to download and install at least the free Dropbox client on to all your computers.