As a freemium business, Dropbox knows that its users find value in incremental free storage for their accounts. Dropbox also knows that, while its product is hugely popular, it is not inherently viral (in part because it's not a social or entirely web-based product).
By leveraging their userbase, adding simple incentives and integrating with other social platforms (namely Facebook, Twitter, Gmail, etc) - Dropbox has implemented a killer viral marketing program... And in a way, it is as much social as it a traditional affiliate program. Each link is encoded with a specific 'affiliate' ID that tracks whether friends register for Dropbox accounts; and if they do, your account is upgraded.
Very simple and apparently very effective: in one post on Facebook, I drove two registrations.
Also worth noting, this implementation doesn't directly integrate Facebook Connect, but it certainly could. Dropbox could integrate users' Facebook 'address book' and individually invite each person, send custom messages, etc.