These piezo gyros have a tiny little tuning-fork-like thingie in them that vibrates in a vertical plane. As the heli yaws, the vibration of the fork wants to stay in the same plane and produces a sideways force that is proportional to the rate of yaw. The tiny computer in the gyro senses this sideways force and decides how to push the rudder servo based on how you have the gyro set (mode, gain, etc.)
The gain control is like a volume or sensitivity control for this system.
Newer more fancy gyros like the JR ring gyro also use a silicon micromachine, but in a ring configuration, and the vibration of the ring is like a wineglass "singing" when you drag a wet finger around the edge. The nodes of the vibration stay oriented in one direction, as the heli yaws around the electronics sense the "precession of the nodes" and the computer does the same thing with the info as the other gyros.

The difference in modes has to do with what you want the gyro to do. If you are interested in having it hold a heading until you command a yaw, it behaves a little differently than a rate mode gyro, which is designed to give even yaw rates but will allow the heli to streamline into the relative wind.