Above about 10,000 feet people need help to breathe. Pressurized aircraft pump air into a sealed cabin and let it out through an outflow valve; the controller moves the valve to set the cabin altitude (the altitude the cabin pressure matches), usually no higher than 8,000 feet in airliners. Closing the outflow valve raises cabin pressure; opening it lowers it. The differential pressure is cabin pressure minus outside pressure, and the structure has a maximum.
Safety (pressure relief) valve: opens if the differential gets too high.
Negative pressure relief valve: stops the outside pressure getting higher than the cabin's, for example in a fast descent.
Dump valve: lets all the pressure out, for landing or an emergency.
The air comes from engine bleed air, a turbocharger's compressor, or electric compressors.
Air conditioning and heat
An air cycle machine cools hot bleed air with heat exchangers, then lets it do work spinning a turbine, which drops its temperature sharply. A vapor cycle system works like a home refrigerator: a compressor, condenser, expansion valve and evaporator move a refrigerant (often R-134a). Small aircraft heat the cabin with a shroud around the exhaust muffler; a crack there can put carbon monoxide in the cabin, so mufflers and shrouds are inspected closely.
Oxygen
Use only aviator's breathing oxygen, which is very dry so moisture can't freeze in valves and lines. Systems are continuous flow (simple, for light aircraft), diluter demand (mixes cabin air in, flows only when you breathe) or pressure demand (for high altitude). Many airliner passenger masks use chemical oxygen generators, which get very hot when started and are hazardous to ship.
Keep oil and grease away from oxygen fittings: under pressure they can ignite on their own. Use only approved leak-check solution and thread lubricants, and no smoking or sparks nearby.
Practice
In a pressurized aircraft, what happens to the cabin when the outflow valve closes further?
More info
Quick reference
Cabin altitude
The altitude the cabin pressure matches.
Differential pressure
Cabin pressure minus outside pressure.
Outflow valve
Sets cabin pressure by how much air it lets out.
Air cycle machine
Cools bleed air by expanding it through a turbine.