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:
Let's look at the ECS for the Airbus A320.
1. Location of elements
A view of a pack with the triangular ram air inlet (the ram air outlet has been removed):
The pack includes:
3. Air cycle machine
4. Pack operation
A pack works this way:
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
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.
There are 4 phases to an internal combustion engine, even a gas turbine: suck, squeeze, bang, blow.
Bleed air is taken out of the engine after squeeze but before bang