I teach Nitrox.
Lots of misinformation and opinions on this post.
Nitrox is the generic term for Enriched Air Nitrox, or EAN. Basically, you are breathing a gas mixture that has a higher percentage of oxygen in the mix and, by definition (and physics of gas laws) a lower percentage of nitrogen.
As you recall from your diving class, "air" is basically 21% oxygen and 79% nitrogen. At sea level, this is a percentage and ratio.
As you dive, the pressure at depth increases linearly with depth. The total pressure your body experiences is the sum of the partial pressures for each gas. The percentages in your system do not change with depth (assuming breathing off a single cylinder, and doing a no-decompression dive). But as you go deeper, you have an extra atmosphere of pressure for every 33 feet you descend, which is added to the pressure at the surface.
For general recreational diving, your breath a tank of "air". You recall from your dive tables and/or computer class, that you have limits of depth. This is due to the partial pressure of nitrogen in your system, coupled with your bodies ability to off-gas this nitrogen, within a given amount of time, for a given depth. You recall the "residual nitrogen", which is the excess your body is not able to quicly off-gas. It takes time to get rid of that quantity out of your tissues. The off-gassing is done at a certain rate. So it takes a certain amount of time to get of "X" quantity of excess nitrogen.
Logically, you can't really "breathe faster" to get rid of the excess nitrogen, once you're on the surface. It takes time to get it out of the tissues, and there are a lot of variables in tissues.
So, if you have a certain rate, it takes a certain time to get of a quantity. If you can't change to rate, but want to reduce the time, logically you would have to reduce the quantity. That is what Nitrox does - it reduces the quantity of nitrogen in your system by reducing it in the breathing gas.
For a standard Nitrox mix of 32% oxygen, that means the nitrogen goes from 79%, down to 68%. This, in turn, means you have less nitrogen molecules in your blood stream. So if you have less nitrogen molecules, it takes less time to get rid of them. This means reduced surface interval time.
On the flip side, breathing Nitrox actually reduces the maximum depth of your dive. This is due to oxygen being toxic above a certain partial pressure. Without getting too technical, think of a pie that has not been cut. Think of the whole pie as one atmosphere of pressure. If you cut the pie into two pieces, one 79% of the pie, and the other 21% of the pie (the percentages of nitrogen and oxygen in air), you have one "atmosphere". Okay, each of those pieces are the "partial pressure" of the gasses, that make up the whole pie (1 atmosphere). So when you go diving, you have one "pie" at the surface. You go down 33 feet, and they hand you another "pie" of pressure. For every 33 feet of depth, you get a "pie" of pressure. So far so good. So at the surface you have 0.79 piece of nitrogen, and 0.21 piece of oxygen. Every 33 feet, the pieces increase. For instance, at 66 feet, you would have 3 x each piece, or 3 x 0.79 nitrogen, and 3 x 0.21 oxygen, totalling 2.37 "pieces" of nitrogen, and 0.63 "pieces" of oxygen. These "pieces" are simply the partial pressures of each gas, at that depth.
Studies have found that you should not exceed 1.6 "pieces" of oxygen, on the high side, and better to keep around 1.4 total "pieces" of oxygen. If you took a tank of air, at 1.4 PPO (partial pressure of oxygen), this equates to 1.4 / 0.21 = 6.66 total atmospheres ambient pressure. Subtracting out 1 atmosphere at the surface - 5.66 total atmospheres, or 187 feet of depth (you would be seriously narced at that depth, and some dive groups will let you "spike" a dive for 30 seconds, with twin tanks and safety divers).
But beyond the 1.4 PPO, too much oxygen is toxic for the Central Nervous System.
But with Nitrox, at 32%, your total gas mix is 0.32 oxygen, and 0.62 nitrogen ratio. If you take 1.4PPO divided by 0.32, you get 4.375 total. Subtracting out 1 for ambient pressure at the surface gives you 3.375, times 33 feet/ATM = 111.375 feet. This is what is called the MOD, or maximum operating depth for a 32% EAN mix.
So what is the advantage, if it costs more?
Several.
Allows extended bottom times at moderate depths (look at the charts for both mixes, and "run the numbers").
Reduces the feeling of fatigue in repetive diving (live aboards, or places like Bonaire where you can dive 5-6 times/day) (no "scuba nap").
Reduced surface interval.
Reduces incidence of nitrogen narcosis.
The time to sit in a course takes about 2 hours. All of the above is discussed in great detail, as well as analyzing a cylinder. I wish I had done it right after I had been certified.
On the gear, if you have your own gear, everything needs to be O2 cleaned. This simply means impurities are removed, and some of your O-rings may need to be switched out. You don't use silicone lubricants any more, after switching over (tends to oxidize, with resulting off gas being toxic).
Hope this helps.