In an automatic transmission, the crankshaft of the engine turns a torque convertor. The outer shell of the torque convertor is the rotor; and it turns a pump inside the transmission which circulates and pressurizes hydraulic oil (transmission fluid). The fluid acts as a lubricant, a power transmission medium, and to actuate various clutches to facilitate shifting gears. In this description, I will use "oil" and "transmission fluid" interchangeably; transmission fluid is just oil of a certain viscosity, with special additives, anyway.
The torque convertor is filled with oil. As the engine turns, vanes on the rotor cause oil to flow inside the housing. At idle, or below a certain speed (known as the "stall speed") the oil moves like a liquid and doesn't transfer all that much power to the stator, which is connected to the transmission's input shaft. But when you step on the gas, as the engine RPM rises, the viscosity of the oil resists the movement of the rotor. Since the rotor must turn, the oil moves with it, and flows against the stator. At a high enough speed--the stall speed--the stator rotates at the same speed as the rotor.
Of course the torque convertor is always transmitting some power to the transmission. This is why the car moves when it's in gear and you just take your foot off the brake. But the connection, being a viscous one, is not a solid one, even when the torque convertor has "stalled"; there is always a little slippage. This is one reason automatic transmissions are sometimes referred to as "slush boxes".
The power then flows through a set planetary gears, and which part of the planetary gear system is stopped determines what gear the transmission is in. This is controlled by the valve body, which is essentially a sort of fluidic computer. It integrates the variables of input speed, output speed, throttle position, and fluid pressure, and determines which gear should be engaged.
Modern cars don't have a complicated valve system; instead the gears are selected by the car's powertrain control computer.
Cars of relatively recent manufacture may be equipped with a lockup torque convertor. This is a torque convertor which has a clutch in it; above certain speeds, the clutch is engaged, providing a direct mechanical link between engine and wheels. This allows the car to have a torque convertor which slips more, for better acceleration, without giving up high-speed economy; above a certain speed, the car's engine computer locks the torque convertor clutch.
In general, a torque convertor which slips a bit provides better performance; but of course this occurs at the expense of fuel economy.
In the so-called "auto-stick" transmissions, the driver has the option of placing the gearshift lever in a special gate which allows him to bump the transmission into higher or lower gears, as he sees fit. There is no clutch, and the transmission still has a torque convertor. (The torque convertor may have a lockup clutch in it, but that doesn't count.)
These transmissions essentially combine the worst features of automatic and manual transmissions.
People do not generally buy cars with manual transmissions because they enjoy throwing the gearshift around; they buy them because manual transmissions have certain performance advantages over automatics.
With a manual transmission, there is a direct mechanical link between engine and wheels, whenever the car is in gear and the clutch is engaged (ie no foot on the clutch pedal). The driver has very precise control over the car's speed and handling characteristics. All of the engine's torque is sent to the transmission, and none is lost pumping transmission fluid or spinning a heavy torque convertor.
With an automatic transmission, the vehicle will generally have a higher towing limit--the torque convertor acts to multiply the tractive effort available--and of course the largest benefit is ease of use. You put the transmission into "drive" and forget about it until you reach your destination.
Besides, a skilled driver doesn't need a special shift lever to shift gears manually with an automatic transmission. You just use the other positions on the shifter--1, 2, etc. Besides, most experienced drivers can control an automatic transmission's upshift with just the accelerator.
So why build the thing that way?
Emissions. Manual transmission cars give off a little "burp" of nitrogen compounds when the engine is suddenly unloaded, during shifting. Automatic transmission cars don't do that; the auto trans provides a steady load with little variation. Automakers can make their cars a lot "greener" if drivers don't insist on banging gears.
Besides that, it's cheaper to build cars if you don't have to install the plumbing for clutch lines and such on some of them. Fewer variants means more profit, always.
Anyone who tries to sell me an "auto-stick" car by trying to convince me that it's for "performance" is either going to lose a customer, or get his face laughed in.