To add to MoneyMon's post, another possibility is if you traded an outside (non-DVC) timeshare into a DVC property through the RCI network.
In that case, you have to get the confirmation number from RCI and then call DVC Member Services (only number I could find is 407-566-3800).
Member Services can then take the RCI confirmation and give you the corresponding Disney reservation number for you to use in all further communication.
In either case (trading in through RCI or MontyMon's example of a DVC trade), you have to add the Dining Plan at least a few days before arrival, but you won't have to pay for it until you check in.
And there's no way to get Free Dining in either of these examples. (The only way to get Free Dining while staying in a villa would be to make a cash reservation directly through Disney)
-Rob
