Keeping it simple here [and with "poetic license" as ther are other valve types] but this will give you the basics. While valves for air and oil are very different to specify and but they work on the same principals BUT we use them in different ways and tend to use different types in each application.
So I'll do Hydro first then outline differences for air.
Your "[glossary]pump[/glossary]" is actually a tank/pump/motor all in one unit. The connection are made via the center [usually Ally] block.
Take "pump" in the diagram as being your pressure port in the block, to which you screw a check valve [CV in the diagram]. Then you screw a T piece to the check, one side of the T goes to the [glossary]cylinder[/glossary] [or cylinders]. [load in the diagram is the weight of the car].
When you hit the lift switch the pump spins and push pressurised oilo throuhg the check valve to the cylinder so extending the rod and lifting your corner. The oil can also go to the "[glossary]dump[/glossary]" [the load of symbols in the diagram !]. You'll see that where the oil goes to the "dump" it pushes on a ball. This ball [sitting on a V which is actually a seat] forms another check valve so the oil can't go back to the tank. When you let go of the switch oil is trapped in the cylinder between the two check valves so the corner stays at the height it is.
[wake up in the back there

]
When you hit the drop switch the "dump coil" moves the valve internals so that the oil is now pushing on the ball and V facing the opposite way, this means that the wieght of your car pushes that ball off its seat so the oil can flow back to the tank. When you let go of the switch a spring pushes the internals back to the starting point.
If you think about how that all works it is fairly easy to see that if you add another dump AND another check valve before it you now control two cylinders. It also shows you why you can't lift corners seperately with one pump but you can drop them seperately.. think about it

In an air system we dont need to return the air to the tank... nor do we want to as the tank is our pressure source [like the pump in hydros] so we dont want to put low pressure air back into it.
Air valves are commonly spool type valves [see any cheap AIM system]. These are OK but ideally need to have a lubrication source in the air circiut which is rairly fitted in our applications. This is why they stick after a time. They work pretty much as the hydro ones decsribed above.
The High flow valves [like the ones used by Easy Street or AirRide Tech] are pilot operated diaphram valves which need no lube and can flow much higher volumes.
Most air systems use no check valves [except for a standard one in the compressor head], just a series of valves [simple open or closed valves] tee'd together to allow air into the bag or out to atmosphere.