I've never heard a cogent explanation (but I bet you could give one ...)
of why these cars don't just run the wheels completely with electric
motors which are powered in turn by a fossil-fuel engine driving a
generator, much like a diesel-electric locomotive. And some batteries
to allow plug-in use. Seems a lot simpler.
I love a challenge, but I've already had that explained to me. John1701 and Hobbit at PriusChat.com have spent way too much of their lives reverse-engineering the car.
You could do a series drive. Mechanically simpler, although current battery technology doesn't support much of a plug-in range or battery lifecycle in a power density that fits the Prius. (50-100 miles, ~18-month replacements.) Despite a plug-in's cheap recharge, batteries just don't seem cost-effective in a consumer-friendly vehicle. But they're close.
But a parallel drive is more efficient. As another poster mentioned, the Prius runs the internal-combustion engine (ICE) in parallel with a motor generator (MG) on a common axis. The ICE does most of the work (which is more efficient without a plug-in) but the MG smooths out the power curve (charging or discharging the battery) and helps the engine with acceleration. (The MG also handles regenerative braking-- no brake shoes are applied unless you really stomp on the brake pedal or until you're below 7 mph.) The ICE operates when needed, most of the time at its most fuel-efficient RPM, and shuts off as soon as it's not needed. The MG fills in the acceleration/deceleration until the engine catches up. The result is a higher power density with no idling wasting gas.
I can't call it simpler. Depending on who's counting the car has over a dozen microprocessors running the accelerator, brakes, battery-charging circuits, MG/ICE, and even the air conditioning. But the design has eliminated a ton of mechanical linkages that were always breaking on our other cars, and the digital cruise control has zero analog-system lag. I'd have to check, but I'm not sure that the car even has an accelerator linkage/cable.
It's not perfect. It's overpriced due to demand, and resale values are ridiculously high. Dealers are generally smug and unsympathetic to customer concerns. The rear-window visibility is oxymoronic. The rear-seat headrests can block the driver's rear-fender blind spots, and they scared the heck out of me when I first checked my peripheral vision for a lane change. The front window needs a lot more tint for sunny weather. The windscreen cover has to be custom-ordered or cut from an oversized roll of reflective foil insulation. The oilpan is exposed to everything sticking up from the road. Tires have a reputation for wearing out more quickly. If you damage the touch-screen display then it's $500-$1000 and it's most inconvenient to attempt to operate the car without it. Lost/damaged keyfobs cost $125-$250 to replace. The antenna sits right where I want to put my longboard roof rack. The rear speakers are muffled by their rear-door panel location. The front speakers are practically in your face on the front door posts.
It's not protected against the silliest common mistakes-- if you accidentally switch the jumper-cable polarity then you'll fry the $5K DC/AC inverter. If you tow the car from behind (with the front wheels on the ground) then similar electrical/mechanical damage can occur. The smart-key system makes it too easy to walk away from a running car with the key in your pocket, especially if you've just switched drivers.
Although the car has a very good warranty at face value, Toyota seems to be very stingy about warranty work and quick to blame mechanics for errors.
But other than that, I'm happy!