If you don't want to wade through the 200+ posts in that ACM pissing contest above, I will give you a more condensed version as it applies to the 737-700, which has two packs (one on each side) located in the A/C bays on the belly of the aircraft forward of the main gear wheel wells.
It's important to note that there other important components that work with the pack, in order to provide temperature control.
First of all, you must have a bleed air source (either from your engines or APU)and a means of control.
When you turn on a pack by placing its respective switch in either AUTO or HIGH, AC
power is supplied to the air conditioning accessory unit (ACAU), located in the E&E compartment between the nose gear wheel well and forward cargo compartment. The ACAU contains the relays that control pack operation once you turn it on. The ACAU opens the flow control and shutoff valve, allowing bleed air to flow into the pack. It isn't until the air hits the turbine of the ACM that it actually does any "work"; more about that later.
The ACM has a single shaft on which you have a turbine, compressor and impeller fan. The fan helps pull air through the ram air duct (those two openings you see on the belly of the aircraft forward of the main gear wheel wells) and through the heat exchangers of which there are two, primary and secondary. Think of them as the radiator in your car, except they use ram air instead of water to cool the bleed air.
The primary heat exchanger cools the bleed air before it passes into the compressor of the ACM. The compressor increases the pressure of the air (go figure) which also increases the temperature. It then passes through the secondary heat exchanger which cools it down again before entering the turbine of the ACM. The air hits the turbine and "expands" causing the pressure and temperature to rapidly drop. Immediately downstream of the turbine is a water separator which swirls the air through a "sock", causing the condensed water to collect in a drain. This water is then sprayed into the ram air flow.
A 35 degree system prevents the water from freezing in the water separator by using some of the warm air from the primary heat exchanger through a low limit valve when a freezing condition is sensed.
The air mix valve then mixes hot bleed air and cold air from the ACM to provide the desired temperature selected in the cockpit.
Without drawing a picture, it's kind of hard to explain. I hope I helped without getting into too great of detail.
Patrick Bateman is my hero.