So in a hydraulic automatic transmission everything is based on pressure. Note I didn't say non-computer automatic! Hydraulic automatics have an analog computer - the valve body - which is used to make decisions on what gear to be in.
There are 2 basic inputs - road speed, and load. Road speed is represented by the governor pressure. In my old Mercedes, it's about twice the road speed. Load is represented by throttle pressure, and can be created two ways, by manifold vacuum or throttle angle / go pedal position. There's also working pressure, the regulated output from the pump. Think of it like the B+ of the transmission. Working pressure moves the servos inside the transmission, and is also the supply governor and throttle pressure are created from.
Inside the valve body are... valves. Spool valves to be exact. The spool valve looks like a thread spool, so the ends form a seal to the bore the valve is in, but in the middle fluid can flow around the valve.
They look like this. In a basic sense, governor pressure is connected to one end of the bore, and throttle pressure to the other.
Let's consider a car with a 3 speed automatic.
Say you're going down the road at 30 MPH, and this results in 60 PSI governor pressure. You're not pressing the go pedal much, so the throttle pressure is 20 PSI. The spool valve is going to be forced to one end of its bore, connecting passages which send working pressure to the clutches, brakes, or bands which shift the transmission into 3rd gear.
Now you want to go faster and push the pedal 3/4 down. That sends the throttle pressure up to 80 PSI, the spool moves the other direction, and connects passages to send working pressure to the clutches, brakes, or bands which shift the transmission into 2nd gear. But you still want to go faster! So you press the pedal all the way to the stop. This makes the throttle pressure go up to 100 PSI. The spool moves all the way to its other stop, and sends working pressure to the clutches, brakes, or bands which shift the transmission down to 1st gear.
In our imaginary car, the maximum speed it can go in 1st gear is 50 mph. As you accelerate past 50 mph, the governor pressure crosses 100 PSI, overcoming the 100 PSI of throttle pressure, the spool moves towards the middle of the bore, and the transmission is shifted into 2nd. Happy with your speed you relax on the pedal, throttle pressure falls, and the transmission is shifted into 3rd.
There's more to most automatics, of course. More gears and more logic results in a more complex valve body.
Here's a theory of operation for the VW 101 3 speed automatic, used from the 60s until the 90s:
http://www.volkspage.net/technik/ssp/ssp/SSP_8.PDF
It has less than half the parts of the 4 speed automatic in my 91 Mercedes
-J