How can power lost after drilling into stud be restored? Is it costly?
It sounds like you drilled into an electrical wire, which will need to be replaced. The amount of difficulty involved in repairing it (and the cost) is mainly going to depend on where the wire is and how easy it is to access. For example, if it was between two outlets that are accessible from a basement or attic, it should be fairly straight-forward. If it was a feeder line running through an exterior wall, probably less so. Fixing it will basically require figuring out what is connected to each end of the wire and running replacement wire.
You should consult an electrician and in the time it takes for him to respond if you have not tripped a breaker or blown a fuse you should immediately go and turn the circuit breaker off or remove the fuse. A cut or drilled into wire can short out and cause a fire. Also an electrician may be able to tell you where the wire runs and how much a repair will cost.