For what it's worth, all an air intake setup can do is lower the resistance of the air entering the intake tract. It cannot add air above atmospheric pressures unless there is some means of actually boosting that pressure.
To explain air density and weight, imagine a column of air as tall as the earth's atmosphere. That column of air weighs in at 14.7 pounds at sea level, so the weight of a charge of air not otherwise boosted by any external means is limited to that 14.7 pounds. That is the force being pushed into the intake tract. Density is the amount of oxygen available in that charge of air to power the burn from the air/fuel mix, and density can change based on temperature of the air. The colder the air the more dense it is, providing more atoms of oxygen to burn.
There are several means of boosting the atmospheric pressure in the intake tract, one of which is ram air (where air pressure generated from a higher air pressure zone at some point on the vehicle body is directed to the air intake),another is a turbocharger, or another means a supercharger.
Additionally, air density can also effect power production, and one can increase air density by keeping or reducing in-coming air charges as cool as possible. Cooler air is more dense and can support a higher charge of fuel, making more power.
A well-designed CAI system will accomplish both of the goals above -- reducing any back pressure that robs the intake tract of the full 14.7 pounds of atmospheric weight (at sea level -- higher elevations have less air weight) and/or keeping the air charge as cool as possible for maximum density of the air charge. A CAI cannot increase air density or weight from atmospheric levels -- it can only preserve all that already exists by not offering resistance (through bends, ripples, etc.) and by not heating the air. An intake can reduce atmospheric pressures by resistance (bends, etc.) so that a couple of psi of air is not available for introduction to the combustion chamber, for instance a restrictive filter that flows less cubic feet per minute than the engine needs causing the weight of the air at the throttle blades to be less than 14.7 pounds (or whatever your air weight is at your elevation).
Additionally, an internal combustion engine will also draw air into the intake tract by way of piston movement. As the pistons go down in their bores, air is pulled into the cylinder. At normal atmospheric levels only the amount of air that equals the volume of the combustion chamber can be drawn into an engine. No more can be introduced than is physically possible by the volume of the chamber. Of course, less than optimal air charges can be introduced my any number of means -- length of time that the valves are open, rpm of the engine (not leaving adequate time to completely fill the cylinder), overlap of valve timing (allowing some air to exit with the exhaust gas), or amount of air-vane opening in the throttle body (or carburetor). It is the regulation of the amount of air entering the combustion chamber that governs engine speed, hence the throttle body plate that limits air flow into the system. Modern engines can also alter fuel charge via computer control of injector pulse width, making for better efficiency.
So, an engine's air needs can be computed very precisely via mathematical formula -- X number of pistons moving at X speed, drawing X amount of air equals the amount of air needed. There is no way to cheat this system with an intake system unless that system can raise the pressure of the air above atmospheric conditions via the means listed above. Our goal then, is to make our intake tract as EFFICIENT as possible so that the maximum amount of cool air enters the intake tract as can in concert with atmospheric pressures. We find CAI systems that do exactly that with some degree of success depending on the individual system, how well it flows (most will flow more than the engine needs) and how cool the air remains in that system until it meets the spark that ignites it.
A secondary (but very important) element of the CAI system is good filtration of the incoming air. Best, of course, would be a ram air stack (like drag racers of old) poking up through the hood of the car with no filter in place -- that would flow the maximum amount of air on every intake charge, but the engine would not last long at all! In the real world, we need to compromise somewhat and find a system that allows as cool of air as possible to get filtered as good as possible and that also feeds as much as possible to the engine whenever it needs it.
Hope this helps to explain what is going on in the intake tract so that good decisions can be made for CAI systems.