The drivetrain begins with pedals which rotate the cranks, which are held in axis by the bottom bracket. Most bicycles use a chain to transmit power to the rear wheel. A very small number of bicycles use a shaft drive to transmit power, or special belts. Hydraulic bicycle transmissions have been built, but they are currently inefficient and complex.
DIY enthusiasts, with tens of thousands of converted bikes using throttle-only, 20+ mph kits, are now officially labeled Moped class. While these bikes handle and pedal-ride just as safely as the class 3 speed pedelecs in many cases, our DIY counterparts will be officially kicked out and left on their own for advocacy and legal acceptance in California. This is a big deal, without a class sticker, any DIY electric bike conversion kit is considered a Moped and not a bicycle.

The oldest patent for an electric bike I've been able to find at the US Patent and Trademark Office is this one, by Ogden Bolton, Jr. of Canton Ohio, which was filed in September 1895 and granted three months later. You can see from these original diagrams that it bears an amazingly close resemblance to modern electric bikes. In the general picture on the left, you can see there's a hub motor on the rear wheel (blue), a battery suspended from the frame (red), and a simple handlebar control to make the thing stop and go. In the more detailed cutaway of the hub motor on the right, you can see there's a six-pole magnet in the center (orange) bolted to the frame and an armature (made from coiled wire, yellow) that rotates around it when the current is switched on. It's quite a hefty motor even by modern standards; Ogdon mentions "a heavy current at low voltage—for instance, to carry one hundred amperes at ten volts." So that's 1000 watts, which is about twice the power of a typical modern bike hub motor.
Electric bicycles use batteries as a source of power and a quiet DC motor as a driving mechanism. On most e-bikes the motor is built directly into the wheel (known as a hub motor) and the batteries are discreetly hidden in the rear rack or frame. Electric bikes can be operated just like normal bicycles, but they can also be power-driven by a throttle or pedaled with the help of pedal-assist (PAS or pedelec).
On the other hand, the battery doesn’t lock into the bike, which means you can’t just leave it on your bike when you park at the bike rack. Anyone could walk by and simply remove your battery. That seems like an oversight to me, though perhaps the designers assumed that such a small battery would just be easy to take with you. And it is. My wife could probably lose this battery in her purse.