Some piston helicopters have a correlation system which is a mechanical way of adjusting power output as the collective is raised and lowered. A governor, on the other hand, uses a closed loop system to sense the RPM and adjust the fuel control to maintain the desired RPM.
A low side governor system failure would be one in which the governor either allows the engine output to go to zero, or to a lower power setting than is required by the current collective pitch angle.
Some helicopters, such as the Robinson, allow the pilot to manually adjust the throttle even when the governor is activated. In this case, the pilot could simply increase the throttle manually by reference to the tachometer, and thus work around the governor failure.
Other helicopters, such as the Bell, have a position on the throttle or flight condition lever in which the governor is in charge of engine power output, and in this position the throttle can not be manipulated by the pilot. For instance, in Bell helicopters, you roll the throttle all the way on to activate the governor. In this position, the throttle is already all the way on, so the pilot has no way of commanding more power than what the governor selects. In a helicopter with that sort of a system, the only choice the pilot would have is to reduce the collective until the proper RPM is attained. This might be at a fairly high power setting or a fairly low one, depending on how the governor has failed. If the governor has failed at a low enough setting, the pilot may have no choice but to autorotate. If the governor has failed at some moderate power setting, it may be possible to maintain powered flight at a lower speed, or with a slow descent.
The Robinson governor can also be overridden in this situation by rolling the throttle to a lower power setting, thus the Robinson can easilly be recovered from a high side failure. Generally the pilot would just turn off the governor in this situation and rely on the throttle correlator (mechanical) to help him maintain proper RPM.
Most turbine helicopters will allow the pilot to retard the engine power output using the throttle or power lever. Thus a high side failure can generally be handled by rolling off throttle or pulling back on the engine condition lever until RPM is restored to the green range. In an extreme situation the pilot might have to turn off the fuel in order to shut down the engine.