In Scuba diving, recreational divers will usually dive on compressed, filtered air, containing approximately 21% oxygen, and 79% nitrogen. The limiting factor to how long you can stay under water on repetitive dives is, perhaps surprisingly, not the amount of air you have in your tank, but the amount of nitrogen you have in your body: At pressure, nitrogen dissolves into your body tissues; too much nitrogen can lead to decompression sickness.
One way of reducing is to dive with a different oxygen/nitrogen blend; and this is what Enhanced Air Nitrox (EAN) is: A gas blend with more oxygen and less nitrogen. Typical blends are Nitrox32 (32% oxygen and 68% Nitrogen), Nitrox36, and Nitrox40. Gas blends of higher than 40% oxygen are rare, because the level of oxygen causes unacceptable risk of combustion and accelerated oxidation of the equipment used to deliver the gas to the diver.
So: The people who dive on Nitrox generally fall in two camps: Those who use Nitrox to further reduce the risk of decompression sickness, and those who are comfortable with the risk, and choose to use Nitrox to stay under the water for longer.
A corollary of Nitrox diving putting less nitrogen into your body, is that you can stay under water at shallower depths for longer: More time spent at pressure with less nitrogen means less nitrogen absorbed into body tissues, so more repetitive dives are possible. This is particularly relevant when you do repetitive dives, such as on a liveaboard dive trip, where you live on board a boat, and do 4-5 dives per day for several days in a row.
Removing nitrogen helps with the risks of decompression sickness, but introduces a new risk: That of oxygen toxicity; the fact that high partial pressures of oxygen can put you at risk of central nervous system issues, including blackouts, convulsions, and optical/aural problems. You don't have to be a dive medic to realise that losing control of your body at depth is a Bad Thing: Drowning being the most obvious problem. The general rule is to stay at a partial pressure of O2 (PO2) of less than 1.4. That means that each gas blend has a maximum operating depth (MOD). When diving on 21% oxygen (i.e. normal compressed air), you can theoretically dive to 57 meters without problems. With 32% oxygen, the max depth is 32m/105ft, and with 36% oxygen, the maximum recommended depth is 28m/92ft. See oxygen toxicity for more information.
There are a wide set of rules and regulations connected with diving on Nitrox, including using either 'air equivalent depth' dive tables, specific nitrox dive tables (which typically only list Nitrox32 and Nitrox36 blends), or re-programming your dive computer to calculate the correct oxygen percentage.
Realistically, all the theory of diving on Nitrox can be summarised into 'Learn how to use your dive computer, use it, and don't be a dick'.
Beyond the theory, diving on Nitrox is exactly like diving whilst breathing compressed air: Breathe normally, and the equipment (and your body) will take care of the rest.