pressurisation is achieved by feeding air into the cabin, and controlling the rate at which its allowed to outflow to atmosphere, the valve which controls this outflow is called an 'outflow valve', usually seen towards the back end of the aircraft on the lower half of the fuselage.
As has already been said. Air is bled off a compressor stage on the engine determined by, power setting/manufacturer, this is hot and high pressure, sent through a pressure regulater and shutoff valve, then through a pre cooler and ducted down too the packs, here it is either sent through an air cycle machine and cooled or bypassed and mixed by a temperature control valve downstream with warm air again to trim the temperature, this mixing is a function of temperature control from the flight deck. This conditioned air is then fed into the zones in the cabin through manifolds, this feeding of air pressurises the cabin, controlling of this pressure is done by modulating the outflow valve to achieve desired cabin pressure.
The outflow valves position is controlled by a cabin pressure controller, this can either be done automatically or manually. the target is too keep the cabin at a nominal 8.9psid(differential), cabin pressure is expressed on aircraft as cabin altittude which is the cabin altitude in relation too sea level, usually around 8,000ft, this will give a safe psid.
Look at it as a flight profile, aircraft starts its flight at the airfield altitude, as the aircraft climbs, the ambient pressure decreases, the higher the altitude the lower the pressure, so the altitude inside the aircraft also increases but at a LOWER rate than the aircraft itself, this is cabin altitude, controlled on climb descent by cabin vertical speed in 1000's Feet per minute, so the outlfow valve, controlled by the CPC
's controls cabin altitude and cabin v/s too maintain a safe cabin psid.