The way they work is pretty simple: The motor kicks in with extra assist when you pedal, and there are different levels of assistance. That's it. Just turn it on, pedal, and go! You can also turn the assist mode off and ride it like a regular bike. And e-bikes stop assisting at a certain speed (in the U.S., it's 20 mph or 28 mph depending on class) for safety, of course.
E-bikes mostly use motors and battery options from a few major suppliers: Bosch, Yamaha, Shimano, and Brose. A few other brands exist, but are less reliable or powerful. Some, like the Yamaha system, have more torque and others are quieter. But generally all four make good options. Look for motor output (in watts) which will give you an idea of total power. But watt hours (Wh) is perhaps a better figure to use—it takes into account battery output and life to give a truer reflection of power.