What is the function of an air filter on a vehicle?

For the engine. Imagine dust getting inside the cylinders and scratching the crap out of everything. Without one your oil would look like mud in a month.

What JP said.The air that goes through the filter gets mixed with the gas when it’s injected into the cylinders, so it has to be free of abrasive particles. Otherwise the movement of the pistons would cause all that grit to wear away at the piston rings and cylinder walls.

There is a separate filter for the air that circulates in the cabin.

@whitetigress yes some vehicles have a separate filter for the inside air in the vehicle as well. My Audi has two filters. One for the engine and one for the inside cabin.

All cars have at least 1 and many cars have 2 air filters.
An “engine air filter” is on every car. The second, less common filter is often called the “cabin air filter” or “pollen filter”.

What @judochop said. An air filter is to filter air, regardless of where it is going. Very likely you have one to filter cabin air, and you definitely have one to filter engine intake air. These filters are mainly to prevent particulates from gunking up precision movements in the engine and stave off respiratory issues. In the bigger picture, there is a range of pore sizes that filters have. Some are designed to just trap large particles, while others are designed to remove bacteria and even viruses. Very often in a ventilation system you will have a sequence of filters from coarse to fine grain to prevent the expensive fine filter from gunking up really fast.

For the engine, gunking up is the least of your problems. The scratches you would get inside the cylinders would have about the same effect as losing your piston rings; oil blow-by, loss of compression, and generally not allowing the pistons to do their thing. This on top of your other issues.

For the cabin, they make the air a little nicer and, more importantly, keep the grit out of your blower motor and your interior. My car lacks one, and you can imagine how dusty my dash is.

Thanks all!