Air is taken from the compressors, then it is cooled by the air that comes from the ram air inlet. After that, it is compressed and cooled again by an expansion turbine. When finally air is condensed, it mixes with bypass bleed air. What is bypass bleed air?
You almost have the answer in your question; bleed air is hot, compressed air taken from the engine compressors. Bypass bleed air here refers to some of that air which is bled off the engine, but not routed through the air cycle machine (pack). When the air comes out of the pack, its really cold. The pilot has a knob which blends the cold air from the pack with the hot bypass air to adjust the temperature that comes through the actual ductwork; this is similar to the blender box in a car.
For example, in this image taken from this answer on SE Aviation, the bleed air from the engine comes in at the top right corner, and you can see the line for bypass bleed air along the top of the diagram in the pipe labelled "hot air".
Bypass bleed air is a fraction of the bleed air coming from the core compressor. It's named like this because it doesn't cross the air cycle machine of the packs as does the regular bleed air.
On the following picture this bypass bleed air is identified as trim air. Its purpose is to be mixed with cooler bleed air from the packs to modulate the temperature in the different zones.
Mixing is done by the mixing and distribution unit (see more below).
Use of trim air on the Airbus A320 family. (Source: A319/A320/A321 ECS Familiarization and Training Manual)
Boeing has open a new approach for air conditioning and pressurization using ram air rather than bleed air. The first aircraft with this approach is the B787, this requires dedicated compressors.
The air conditioning and pressurization systems on large airliners are quite similar. Here is the overall picture:
Air conditioning and a part of the pressurization are obtained using the environmental control system (ECS)
Bleed air, the high pressure air taken from the engine cores and APU, is very hot.
Ram air, the cool low pressure air taken from the aircraft belly, is used to cool the bleed air in heat exchangers. This ram air, which gets warmer in the process, is rejected in the atmosphere without entering the cabin. A fan is used to create the ram air flow when the aircraft is on the ground.
Air conditioning packs are mainly air cycle machines (refreshers) that receive pre-cooled bleed air and deliver air at ambient temperature. There are usually 2 or 3 packs for redundancy.
Some bleed air bypasses the ACM and can be used later by the mixing unit.
Air from the packs goes to the mixing and distribution unit which role is to mix air from packs with air already in the cabin, and deliver the result to different zones of the aircraft.
The unit can vary temperature according to zones by mixing packs output air and bypass bleed air. The unit also mixes air from the ground supply when at the gate and engines and APU are not running.
Air in the cabin is recirculated by two or more fans which return it to the mixing unit.
As air is pumped into the aircraft by the mixing unit, air also exits the aircraft through one or several outflow valve(s). The exact quantity of air leaving the aircraft is controlled so that the pressure inside the aircraft is at the required value.
This value is usually expressed as cabin altitude, that is the altitude we would find this pressure in the standard atmosphere. Pressure is about 1013 hPa at sea level and then decreases continuously when altitude increases.
However the cabin altitude doesn't climb as high as the aircraft because there would be an insufficient oxygen pressure in the cabin. The cabin altitude is kept proportional to the aircraft altitude, but usually the minimum of pressure is a value equivalent to 6,000 to 6,500 ft.
All those elements are driven by controllers that can be monitored and programmed by the crew.
Let's look at the ECS for the Airbus A320.
1. Location of elements
(When not specified, the source of the images is the A319/A320/A321 ECS Familiarization and Training Manual)
A view of a pack with the triangular ram air inlet (the ram air outlet has been removed):
The pack includes:
Two air-to-air heat exchangers: main (MHX) and primary (PHX). Ram air is used as the heat receiver fluid for both.
A centrifugal compressor (COMP) and a turbine (TURB). They compose the air cycle machine (ACM) which cools and reduces pressure of bleed air.
A reheater (RH), a condenser (COND) and a water extractor (WE) to dry bleed air.
3. Air cycle machine
4. Pack operation
A pack works this way:
Hot high pressure bleed air is directed to the PHX which cools it a bit so that it can be processed more efficiently by the ACM.
Cooled air is sent to the compressor. Increasing the pressure allows for more heat to be extracted in the next step. Air gets very hot again.
Compressed hot air is sent to the MHX which remove as much heat as possible, possibly below the ram air temperature, which is great while on the ground.
Cool air, which contains water vapor, goes to a reheater and a condenser to create liquid water, water is collected by the water separator. This water is reinjected into the ram air to cool it before it is used in the exchangers.
Cool air finally rotates the turbine which extracts energy from it to drive the compressor and a ram air fan. The energy loss produces a cool low pressure air which can be used in the cabin and cockpit.
The exact temperature and moisture of air delivered by the pack are obtained by mixing air from the ACM turbine, cooler air from the PHX and warmest bypass bleed air taken upstream the MHX.
Water extraction is controlled by adjusting the temperature, in the same way, of the airflow going through the condenser.
Bypass bleed air is also delivered by the pack to the mixing unit for zone temperature trimming.
5. Water extractor
Collected water is reinjected into ram air flow.
6. Air temperature and pressure at each step
There are 4 phases to an internal combustion engine, even a gas turbine: suck, squeeze, bang, blow.
Suck: Air is sucked into the engine
Squeeze: The air is compressed by the turbine
Bang: The concentrated air is mixed with fuel and ignited.
Blow: The expanded gases from the ignited fuel push out the back of the jet
Bleed air is taken out of the engine after squeeze but before bang
The Environmental Control System (ECS) which supplies air to the cabin usally uses bleed air from the engines in most aircraft. The figure below shows the engine bleed air system in a Boeing 767.
Image from Commercial Airliner Environmental Control System- Engineering Aspects of Cabin Air Quality by Elwood H. Hunt et. al, from Boeing
The main air supply for the ECS usually comes from the compressor, where air is taken from high/low pressure areas. These are used during different phases of flight.
The air is sometimes taken from the bypass air of the engines for the purposes of cooling. As the air is compressed in the compressor, the temperature of the bleed air rises (especially the one from HP stage) and the bypass air is used to cool it.
These answers are all correct, however. It should be noted that there are 2 separate types of bypass bleed air involved in this air conditioning system. There is bypass air taken just downstream of the flow control valve at the entry to the pack system that goes straight to the trim air for cabin temp control. The other bypass bleed air is involved with preventing ice buildup in the condenser/water extractor. this is the air that flows through the bypass valve(BPV), bypassing the compressor and main heat exchanger to provide hot air downstream of the turbine. The valve opens based on difference in pressure due.