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.
Vehicles with carbureted or older fuel injected engines normally have the air filter located right on top of the engine under a cover that can be removed by undoing a wingnut or two, or a couple of clips. Once the cover is removed the air filter can be lifted out. Remove any large debris from around the filter and then use a vacuum to suck up any smaller debris. To check the air filter hold it up to the sun or a bright light. You should be able to see light through it, otherwise it should be replaced. Inspect the filter to ensure that there are no tears and that it will seal properly. If it still looks good re-install the filter and then replace the air cleaner cover and tighten the nuts or clips.
Newer vehicles normally have the air filter located in a box near one of the front corners of the car, attached to the engine via an air intake tube. The cover of the box is normally removed by unclasping one or more clips. Move the cover out of the way, remove the filter, and vacuum the bottom of the box. Check the air filter's condition by holding it up to light and inspecting it. Carefully look for rips or tears in the filter. If the filter doesn't need to be replaced you can re-install it, reposition the air box cover, and then reattach the clips.
Once the filter has been reinstalled make sure that you haven't left any tools or parts around the engine bay, and start the car. Listen for any obvious air leaks around the filter to ensure that the air cleaner is properly sealed, and then take the car for a drive.