Hydraulically operated clutches work in much the same way as hydraulic brakes. There is a master cylinder, a line, a slave cylinder, a push rod, and a clutch basket. Typically these systems are easier to operate and can push a heftier clutch around, so they are used on bigger bikes.
Problems occur in the same fashion as failed brake lines. You can get leaks, infiltration of water or sediment or rust, or faulty master or slave cylinder components, requiring cleaning or replacement. If your clutch lever is feeling mushy without the usual amount of resistance, or if you are having a harder time shifting or getting back into neutral, it may be time to service these parts. Sometimes all you need to do is flush the line, which is done in the same way as the brake line.