Oxygen concentrators work on the principle of ‘rapid pressure swing adsorption’ which is where the Nitrogen is removed from the air using zeolite minerals which adsorb the Nitrogen, leaving the other gases to pass through and leaving oxygen as the primary gas. Once the oxygen is collected the pressure then drops which allows Nitrogen to desorb and be expelled back into the air.
Acetone-3D-vdW
An oxygen concentrator has an air compressor, two cylinders filled with zeolite pellets, a pressure equalizing reservoir and valves and tubes. During the first half-cycle the first cylinder receives air from the compressor, which lasts about 3 seconds. During that time the pressure in the first cylinder rises from atmospheric to a few times normal atmospheric pressure (about 20 psi) and the zeolite becomes saturated with nitrogen. As the first cylinder reaches near pure oxygen (there are small amounts of argon, CO2, water vapour, radon and other minor atmospheric components) a valve opens and the oxygen enriched gas flows to the pressure equalizing reservoir, which connects to the patient’s oxygen hose. At the end of the first half of the cycle, the air from the compressor is directed to the 2nd cylinder. Pressure in the first cylinder drops as the enriched oxygen moves into the reservoir, allowing the nitrogen to be desorbed back into gas. Part way through the second half of the cycle there is another valve position change to vent the gas in the first cylinder back into the ambient atmosphere, keeping the concentration of oxygen in the pressure equalizing reservoir from falling below about 90%. The pressure in the hose delivering oxygen from the equalizing reservoir is kept steady by a pressure reducing valve.
Portable oxygen concentrators
These have been around for decades, but older models were bulky, unreliable, and were not allowed on airplanes. Since 2000, manufacturers have improved their reliability and size and they now produce 1-6 lpm of oxygen. The portable concentrators plug directly into a regular house outlet for charging at home or hotel, but they came with a power adapter that can usually be plugged into a vehicle DC adapter. They are able to operate from the battery power as well for either ambulatory use, or away from a power source, or on an airplane.
Portable oxygen concentrators operate on the same principle as a home domestic concentrator, operating through a series of cycles. Air passes from the miniaturised air compressor and through the molecular sieve of zeolite granules, which adsorb the nitrogen. Some of the oxygen produced is delivered to the patient and some is fed back into the sieves to clear them of the accumulated nitrogen, preparing them for the next cycle. Through this process, the system is capable of producing oxygen of up to 90% consistently. The latest models can be powered from mains electricity supply, 12v DC (car/boat etc.), and battery packs making the patient free from relying on using cylinders & other current solutions that put a restriction on their activities and mobility due to  time, weight, and size.
Most of the current portable oxygen concentrator systems provide oxygen on a pulse (on-demand) delivery in order to maximise the purity of the oxygen. The system supplies a high concentration of oxygen and is used with a nasal cannula to channel oxygen from the concentrator to the patient.
References: http://en.wikipedia.org and http://www.inogen.com and http://hme-business.com