Used to be all cars were stick shift, auto's arriving in the 1950s.
Basically, a torqe converter bolts to the engine flywheel where a clutch would go. It has vanes in it like a turbo. the spinning engine spins the vanes moving the fluid. also inside the torque converter are other vanes that are attached to the trans input shaft - as the revs come up the engine driven vanes move the fluid and that pushes the trans shaft vanes into moving. That'll get you 1 gear and the ability to stop in 'drive'.

For more gears, there are many more parts, clutches (like on a motorcycle) and planetoary gears, pressure regulators, etc. I am not an expert on teh insides of these things!

Older chechanical trannies shifted when pressure of the fluid reached certain points, also either vacuum from the engine, kick down linkages or on 80s GMs, a Throttle valve cable would control shift points. Starting in the late 80s, electronically shifted transmissions appeard, where the computer activates a solenoid valve in the trans to shift gears.

The manual automatics use those electronic solenoids to allow 'manual' shifting. The newer european cars are now using manual transmissions with electronically actuated clucthes to behave like real automatic transmissions.

Which is better? Depends... automatics have come so far that they are probably the better choice in most situations. With lock up torque converters and electronically controlled shifts the efficiency is as good or better than a manual, and autos shift quicker when drag racing, and certainly more consistently, and can be built to take 1000 HP.

For more info seach for the TCI and B&M websites. Also, Car Craft magazine this month has a couple of articles on torque converters and Gm's 700R4 trannie.