Mutually private computation
Imagine the ideal protocol. It would have the most trustworthy third party imaginable — a deity who is on everybody's side. All the parties would send their input to this god. God would reliably determine the results and return the output. In addition, God, being the ultimate in confessional discretion, would ensure that no party would learn anything more about the other party's input than they could learn from their own input and the output.
Alas, in our temporal world we deal with humans rather than deities. Yet, too often we are forced to treat people in a nearly god-like manner because our infrastructure lacks the security needed to protect ourselves.
To an astonishing extent, network security theorists have recently solved this problem. They have developed protocols that create virtual machines between two or more parties. Multi-party secure computation allows any number of parties to share a computation, each learning only what can be inferred from their own input and the output of the computation. These virtual machines have the exciting property that each party's input is held in strict confidence from the other parties. The program and the output are shared by the parties.
For example, we could run a spreadsheet across the Internet on this virtual computer. We would agree on a set of formulas and set up the virtual computer with these formulas. Each participant would have their own input cells, which remain blank on the other participants' computers. The participants share output cell(s). Each participant inputs their own private data into their input cells. Alice could only learn as much about the other participants' input cells as she could infer from her own inputs and outputs.
I go on to briefly describe applications such as confidential auditing and auctions with private bids, here.