Shinra Technologies was a cloud computing service from Square Enix. Unlike OnLive and PlayStation Now, it wasn't intending to offer existing titles to incompatible devices by hooking input and returning video. They wanted to use this service to create new titles, even with third-parties like Ubisoft, that offload big computational elements to their servers. Those should be latency-insensitive parts, but details would be game dependent of course. This is similar to Microsoft's Xbox program, which is said to power the upcoming game for the Xbox One, Crackdown 3.
Image Credit: Final Fantasy Wikia
We now say was, because Square Enix killed the program. It was created as a subsidiary to keep it separate from the game development division, which might have given some comfort to third-party developers. It also allowed them to secure funding for the program independently, except that investors did not sign on. Without external capital, Square Enix dissolved the division at a loss of about $17 million USD.