The horsepower an engine can produce is related to how much fuel it can burn. To properly burn the fuel you must have a ratio of 14.7 parts air to 1 part fuel. Since fuel can easily be pressurized and forced into the combustion chamber, an engine is extremely dependent on its ability to flow large quantities of air in a short amount of time. To express it simply, the more air you can get into the combustion chamber, the more fuel you can put in and make more power. In most engines the piston's movement to the bottom of the combustion chamber creates a vacuum, drawing in more air. Supercharging is another term for forced induction, meaning cramming more air into the combustion chamber. Turbocharging is a specific method of supercharging, and turbochargers are usually not referred to as superchargers.
Superchargers are typically run off an engine's crankshaft and have some form of pump that produces a pressure several psi above the ambient pressure. This pressure is referred to as boost. Superchargers tend to be most effective at low RPM and run in direct relation to engine speed. With a 1.1:1 pulley on a supercharger it will run at 5500 rpm when the engine is at 5000 rpm.
Turbochargers have a turbine assembly that is turned by exhaust gas exiting the combustion chamber. This turns the centrifugal assembly on the front half of the turbocharger, producing boost. Boost in turbochargers increases exponentially with RPM as they run in direct relation to exhaust pressure. When initially hitting the throttle there is very little exhaust pressure and little boost available. At high RPM there is a plenty of exhaust pressure, which leads to plenty of boost.
Nitrous oxide injection also produces power by introducing more oxygen, but the mechanics of it are far different. Instead of pushing through more ambient air (which is roughly 1/5 oxygen), nitrous systems temporarily inject an oxygen bearing gaseous compound (nitrous oxide, N2O) into the combustion chamber. The nitrous oxide is kept in a bottle, usually located in the trunk. When the driver determines more power will be needed a valve on the bottle is opened either manually or with a solenoid. At the precise moment the nitrous is needed the driver hits a switch. A solenoid near the injectors opens and allows pressurized nitrous oxide to spray into the intake tract. As the gaseous nitrous oxide enters the intake tract its pressure drops causing a drop in temperature. This temperature drop also causes the air in the intake tract to cool and contract allowing more ambient air to enter the combustion chamber.
Nitrous oxide systems require a solenoid dedicated to injecting more fuel. In supercharged applications, the extra air entering the intake draws more fuel into the combustion chamber. With nitrous oxide the additional oxygen is injected after the fuel system has metered the proper amount of fuel. To prevent an overly lean fuel mixture that could rapidly oxidize (burn) vital engine components, a solenoid is opened allowing more fuel into the engine.
Which system a driver chooses depends on their driving habits, money on hand, and available expertise. Installing a supercharger is both expensive and hard work. Installing a turbocharger surpasses a supercharger in both cost and difficulty. Nitrous oxide is the cheapest and easiest to install, but each fill of the bottle typically lasts less than half a minute and costs several dollars.