The engine air filter stops airborne contaminants from getting sucked into your vehicles' engine before it can do damage to the engine.
If your air filter gets too dirty or clogged, your engine won't be able to suck enough air into the combustion chambers. The engine will then run rich (too much gas and not enough air). When this happens, your vehicle may lose power and the engine may run roughly. Additionally, your "Check Engine" warning light may come on.
Your vehicle may stop running altogether if you really neglect the air filter for a long time. Permanent damage is unlikely, but it is possible. Sufficiently dirty or damaged air filters would allow harmful, abrasive contaminants into the combustion chamber.