Power, when activated by a throttle on the handlebar (power-on-demand) or in response to your pedaling (ped-elec), gives you an immediate, nearly silent push. When you release the throttle (or stop pedaling), the motor coasts or "freewheels" - like when you stop pedaling a regular bike. Standard bicycle hand brakes and gearing round out the controls.
"Power-on-demand" means just that - no pedaling required! Although all electric (or "electric-assist") bikes are designed to work with your pedaling, power-on-demand allows you to break the rule. Most systems offer a variable speed control, although some are simply on/off (like the yellow switch pictured). A "ped-elec" wont deliver motor power unless it senses you are pedaling. And its "power output to pedal pressure" ratio is often adjustable. Most people find 400-watt motors adequate for their needs, although folks with steep hills may want more power. Some bikes offer through-the-gearing power assist - i.e. the force of the motor goes through the bikes gearing system - which provides better hill-climbing and top-end speed than direct drive systems with motors of the same wattage rating.