Old automatic transmissions were shifted entirely using hydraulics, not electronics. For this operation the transmission would generate 3 pressures; line pressure, throttle pressure and governor pressure.
Line pressure is generated with the front pump of the transmission. It is used to actuate clutches and bands.
Throttle pressure is a derivative of line pressure. As the name implies throttle pressure is directly related to how far the throttle pedal is pushed. It's a derivative of line pressure because line pressure is tapped to generate it. The tap then goes to a valve which is connected to the throttle pedal. When the pedal is all the way up the valve is open and no pressure is generated. When the pedal is to the metal the valve is closed and the throttle pressure is at it's maximum.
Governor pressure is similar to throttle pressure in that it is a derivative of line pressure but it is generated relative to the speed of the vehicle. Normally the governor is in or near the tail shaft of the transmission. The governor has a set of fly weights (like centripetal advance in a distributor). As you drive faster the weights fly out more closing the valve and raising pressure.
Finally everything comes together. The transmission uses spool valves to shift. When at rest and not on the throttle the governor and throttle pressures are zero and the transmission is in firs gear. The spool valve is held in the first gear position with a spring. Now you take your foot off the break but don't hit the gas. The car will slowly speed up and the governor pressure will go up. When the governor pressure is high enough it will overcome the spring and slide the spool valve over turning off first gear and turning on second gear.
Now suppose that you are already moving in second gear and want to accelerate rapidly. You lay into the throttle pedal which causes the throttle pressure to climb. The spring pressure plus the throttle pressure is now great enough to overcome the governor pressure and slide the spool valve back to turn off second gear and turn of firs gear. You get a downshift.
Even with the engine being computer controlled the first piece of electronics that made it into the transmission is the torque converter clutch (TCC) solenoid. The TCC should only come on when the engine is warm and there is no good way to give that input to the transmission. Since the engine computer already knows the engine temperature, vehicle speed and throttle position it can activate the TCC.
While the engine drivability was constantly improving the transmission lacked something to be desired. The valve body (housing that contains the spool valves) constantly got more complicated with shuttle valves and accumulators and other things to improve the shift quality. Eventually all of the controls of the transmission become electronic.
With electronic controls the vehicle speed and throttle position is read into the computer (engine or transmission depending on the vehicle). The computer then decides when to shift. To shift, the computer activates solenoids to turn gears on and off. This greatly improved the quality of the shifts because the computer can keep track of how the transmission is shifting and then compensate for any shortcomings. The computer pulse width modulates the solenoids and by controlling the duty cycle can change the speed of the gear turning on. This can change how soft or stiff the shift is. Also by monitoring how quickly the gears engage the computer can choose when to turn the gear on. Chrysler calls this monitoring clutch volume index (CVI).