The air filter in your car filters the air going into the engine, removing dust and other debris, slowing engine wear. If the air filter is dirty or clogged it can make your car feel sluggish and hurt fuel economy. When you check your air filter you should also check your car's maintenance guide to see if your car has a cabin air filter that needs to be inspected or replaced as well.
Some vehicles, particularly partial zero-emissions vehicles, have a sealed airbox where the filter cannot be replaced. Because its impossible to visually check the filter on these vehicles they normally include some sort of airflow gauge that indicates when the entire airbox assembly needs to be changed.