OP amp, Attempt Out > Rail
Figure 1. Internals of the ancient 741 opamp. Source: Wikipedia.

Most opamps will have an output arrangement similar to the push-pull arrangement of the old 741. Others will have FET transistors rather than BJTs. In either case if the top transistor (red oval) is turned on the output will be pulled to positive rail. If the bottom transistor (green oval) is turned on the output will be pulled to negative rail. How close they get depends on the exact output configuration and the driving circuitry.


  If you try to obtain a voltage greater than the power rails of an op-amp, what will happen?


The output will turn on as much as it can and will limit your output somewhere between the rails.


  I am assuming that the output == the rail limiting it. Is that correct, or does it break the op-amp?


You are correct in that the output will be limited. It will only "break" it in that if it latches up there may be a short delay for it to unlatch and begin normal operation again.

Note that this does not take into account what happens if the output is driven outside the rails by an external circuit feeding back into the output.