What is an air filter

All vehicles have an air filter because your car requires a mixture of air and fuel to run.
Normal air is full of dirt and dust and if this were to enter your engine unfiltered then it would get damaged over time.

You can run an engine without and air filter, but damage will definitely occur in time and it cannot be repaired cheaply.
 

Your cars original air filter will be either round or panel (flat rectangle) type.

Typically, modern cars with fuel injection use a panel air filter, and those running on a carburettor have a round filter.

These filters normally have a paper core that is folded many 100's of times to give a larger surface area. The core is surrounded by a flexible rubber type outer frame that holds the core in place and forms an airtight seal inside the airbox or filter housing. The reason that the core is folded many times is so that there is a larger surface area for the air to be filtered through. Standard paper filters are very good for filtering out the dirt and dust from the air that the engine is sucking in. They are also cheap and easy to replace.

Whatever the shape, it will sit in the engine bay inside an 'air-box' or filter housing.

This is normally made of plastic and protects the incoming air from the heat of the engine.
It will also have pipework connected to it so as it draws in cool air from the front of the car.
Cool air is denser than warm and is better for fuel efficiency and performance.

NOTES
Regularly changing your air filter will help it to run more fuel efficiently and keep its performance up. It also helps lengthen the life of the engine. Replacement air filters (same as originally fitted) are readily available from most motor-factors and car shops. They are inexpensive and very easy to change. Paper airfilters cannot be cleaned unlike many aftermarket performance filters.

If you want to gain more power from your engine, you should be considering changing
your original panel air filter for an performance induction kit.  These kits replace your
airbox with a cone shaped filter.