Every citizen is issued a unique "wallet" and can use it to vote. The underlying bitcoin infrastructure does need to be presented to the user, but one way that the app could work is that each candidate has a unique numeric identifier associated with him. If Obama's number is 0.05476 and Romney's is 0.02299 then when you click Vote for Obama, the app sends 0.05476 btc to a designated counting wallet. Several methods may be used to count the votes. Either each candidate has its own counting wallet and counting can be done by dividing the total btc in the wallet divided by the unique number, or all of the votes could be mixed into one wallet and the blockchain could be analyzed to determine how many votes of each type were cast. All of the votes are obviously public information on the blockchain, so there can be no funny business. As long as you don't tell anyone your wallet address, your vote is anonymous.
I thought maybe micropayments for content, but I don't think people like to get nickel-and-dimed. Usage would probably be too low to build a real product/business around it. Ideas?
I keep thinking how revolutionary it is yet I can't seem to think of any ideas beyond payments :-(