Disclaimer: These methods do not take into account how dirty or damaged your lenses are.
Toothpaste: This hack is all over the internet. Just apply toothpaste to your headlights and cover thoroughly. Allow it to sit, and then rub in a circular motion to work out the fogginess. Rinse well, and use furniture polish or a headlight protector to prevent fog from happening again.
Sandpaper: Many people swear by using a wet/dry sandpaper. First wet the sandpaper. Then, starting with a coarser grit, work your way to a finer grade as the film disappears. It also helps to clean your headlights with soap and warm water before using this process.
If you decide to use this method, just make sure that you protect your car's paint by applying painters tape around the headlights. Failing to do so could cause more problems for your vehicle if you're not careful.
Lens Repair Kit: If you want a more professional option or if you think the fog on your headlights is heavier than the average vehicle, you can pick up a lens repair kit at your local automotive store. This kit will come with everything you need to complete the job, and runs somewhere around $20.
Insect Repellent: This one surprised me but I guess it actually works! Though it is not a long term solution, and is only about 90% effective, it’s an easy one and you probably have insect repellent laying around your home!
Simply spray the insect repellent on a cloth or towel and rub on your headlights until clear. Then rinse off with soap and water.