How Nitrous Oxide Injection works...  

Liquid Nitrous Oxide along with some additional enrichment fuel is injected into the induction system "somewhere" along its length. Where exactly that position is, isn't important, provided that it goes into the engine evenly, with the same amount of Nitrous Oxide and additional fuel to each cylinder. That's all that's important here -- that it gets into the engines intake where it is carried to the combustion chamber. 

In some cases it is even injected outside the intake system system, aimed towards an open carburettor mouth or throttle body for example. There does not seem to be much difference in the performance and all of the different methods seem to achieve the exact same result.

There are all kinds of daft arguments about "plate" (under Carburettor) systems versus Direct port systems and different kinds of injector or fogger, but in reality it makes little to no difference at all!  The dynamometer has proved this lots of times over many years.  There are sometimes small differences on different engines but which is best is debatable and pretty unimportant in the scheme of things..

Once the extra fuel and Nitrous Oxide gets into the cylinder the heat breaks it down into Oxygen, and Nitrogen during combustion, allowing the now newly released free oxygen to burn the extra fuel. This is where most but not all of the extra power comes from.

Notice that with a correctly set up and running engine there is no additional free oxygen available until AFTER correct ignition at whatever timing point the engine is set up to fire... There is however a greater size of charge in the cylinder during compression, It consists of a much denser charge (Due to the charge cooling effect) and the additional enrichment fuel, along with the engines normal induction charge of air and fuel.  So until ignited by the spark plug the actual mixture consists of a colder, overly rich, (the nitrous has not released its oxygen yet)  denser, larger charge. It is for these reasons Nitrous actually helps prevent detonation in sensible doses of say 30 to 40 percent boost levels.

Nozzles or Injectors

Known as "Foggers" (started by NOS Systems in the US), These inject the Nitrous and use this rapidly expanding liquid Nitrous as it starts to boil off into a gas to atomise or "fog" the fuel as it enters the inlet tract.  The high pressure Nitrous Oxide hits the fuel as it enters the intake, and breaks the fuel into a very fine mist.  The point of injection can be very close to the cylinder for minimal expansion, or at the throttle plate or elsewhere in the induction system. In many cases, depending on how the nozzles are situated and aimed, the normally aspirated airflow will normally increase as well - in part due to the cooling effect increasing the density of all the air/nitrous inside the inlet tract.

Cooling Effects

Cooler intake air is denser and contain more oxygen atoms per cubic foot. So cooler air will allow more fuel to be burned and in turn make more power. A 10 degree drop in temperature can add 1 to 1.5% power to an engine. Nitrous oxide boils at -129°F and it will begin to boil as soon as it is injected. This can cause a 80° or so drop in manifold air temperature. Now if we are dealing with say a 400 hp engine, we can see well over 30 hp gained from the cooling effect alone. This cooling effect also helps the engine deal with and prevent any detonation.
 

Average Power

If you were to build a 550 hp 350 Chevy, it would have to rev to 7000+ rpm to make that kind of power and only make power in a narrow rpm range. A nitrous injected 350 Chevy making 550 hp would make that power at a much lower rpm and higher average horsepower. So the nitrous engine will out perform the normally aspirated engine by a healthy margin.

The reason is that nitrous flow and therefore bhp increase remains constant no matter what rpm the engine is at. 100 bhp extra is still 100bhp extra at 1000 rpm, as it is at 6000rpm!

At lower speeds there is more time for the nitrous to fill the cylinders, so you get more nitrous in the cylinders per power stroke at lower rpm. This will boost power more at low rpm (before the engine is in it's power band). As rpm increases, and gets in the power band of the engine, you will get less nitrous per power stroke, but the engine will start making more normally aspirated power. This really flattens out the torque curve and widens the power band.

So Why Not Pure Oxygen?

Air has only 21% Oxygen, the rest is made up largely of inert Nitrogen. That nitrogen does not aid in combustion at all, but it does absorb energy, expand and carry heat away. When you add Nitrous, it has 36% oxygen with the rest being Nitrogen once the heat of combustion has broken it down. So the more Nitrous Oxide you add, the less percentage of Nitrogen is available to absorb heat. That is why Nitrous increases engine heat very rapidly. If we were to add pure oxygen (which has been tried), the percentage of Nitrogen would fall much faster as more Oxygen was added.

We would not be able to add much Oxygen before heat was a real problem to control. Also compressed Oxygen is in a gaseous form, so adding oxygen takes up more room in the intake system and reduces normally aspirated power, and the amount of Nitrogen from it. By injecting liquid Nitrous, the normally aspirated power only drops slightly and in some cases it increases due to cooling and contraction and it is adding oxygen and Nitrogen at the same time. To put it simply, with nitrous oxide, we can get more Oxygen atoms in the engine and have a lot more Nitrogen as well. Nitrous can make much more power before heat is uncontrollable. And it can be stored as a liquid in pressurised tanks where Oxygen cannot.

See more detail Why we cant use Oxygen!

*Technically the term NOS is incorrect as this refers to an specific company (called Nitrous Oxide Systems, in the US)