Please note that induction kits will only add power if you have a restriction in your air intake. If you haven't tuned it much then the standard airbox is generally fine and up to the job.
We just like cars need air. In an engine the air needs to be matched precisely to the fuel used into what is referred to as an air to fuel ratio or AFR.
For petrol engine cruising at low load an AFR of around 14.5:1 is the ideal, but under load the engine will require more fuel, with typically around a 13:1 AFR. In a turbo engine the AFR might be much higher at around 11 – 11.6:1.
Generally speaking the more air you have the greater the amount of fuel but some air filters can be quite restrictive so we look at the options of upgraded cold air intake. Most engines will automatically trim the fuel to match the air depending on how many unburnt particles there are in the exhaust. 
Adding a cold air induction kit allows a car to suck in air more freely. The cold air induction kit typically comprises a cone or hemishperical filter with no air box and a feed from outside the engine bay or via a vent.
The idea is that more cold air can enter the engine. The standard paper filters that car have in their air boxes are somewhat restrictive to airflow especially if you have uprated the performance of your car.
If the engine can suck in more cold air then it is able to burn more fuel so you are effectively increasing the efficiency and power output of the engine. "Beware – not all engines will benefit from an cold air induction kit! Some will lose low down power."
It is worth noting that not all engines will experience power gains from an cold air induction kit. Smaller engined cars (those under 1.4 especially) will actually feel less powerful with an cold air induction kit. In this case you should use a panel air filter which replaces the standard paper one inside the OEM air box. If you are expecting massive power gains from an induction kit you are likely to be disappointed.
Typical gains on large engines are only around 2-6% and then only at the top end. The main motivation to adding an induction kit is for the distinctive induction roar. In these smaller engines a certain amount of excess oxygen will not be matched with fuel as the injectors are not big enough. This causes the engine to run lean. An air filter should also be matched to a good exhaust system to allow a balanced flow through the engine.
More air in means more exhaust gases. In all open filter induction kits will suffer from under the bonnet temperatures and generally suck in lots of hot air. We all know that hot air carries less oxygen so performance will suffer but it will generally result in better fuel economy. If a car with an open induction kit in the engine bay and no vents is run for a mile or so the under bonnet temperatures will drop to ambient so it is not all bad news.
Removing the airbox makes the intake temperature rise as air is taken from this hot region. Most dyno runs are taken with the bonnet open so will not reflect real world power gains. To combat this problem add a cold air feed to pipe in fresh cold air from outside the engine bay or place a vent near the air intake.