Most aircraft get their pressurized air from the engine bypass air, off a system called the bleed air system (air is bled off of the bypass). Since the bypass air (air that is compressed but not used for combustion, it makes jet engines more efficient) is already pressurized by engine operation it's sort of "free compressed air" and used for a few different purposes. That said, some newer aircraft are choosing to simplify design by eliminating the bleed air system and instead using electrically-driven compressors for cabin air. One advantage this has is that the air doesn't require initial cooling like bleed air does due to the engine bypass being so hot. Mostly, though, it's being done for the reduced cost/complexity and increased safety of eliminating the air lines from the engines.

The air needs to be pressurized so people can breathe normally. The most convenient way to do this is to use bleed air from the turbine compressors, but no-bleed systems (e.g., https://www.boeing.com/commercial/aeromagazine/articles/qtr_...) are possible.