Mix 1/2 tablespoon of cream of tartar for every cup of water. Put as much as you need in your pan to cover all the burn spots. If the burn spots are on the outside of the pan, fill a larger pan with the cream of tartar mixture and place the aluminum pan inside of it.
Simmer the mixture for 10 to 15 minutes. Drain and then rub the burn spots off with a sponge or rag. You can use a nylon bristle brush or scrubbing pad as well.
Mix 1 teaspoon of white vinegar for every cup of water and use this solution to remove the remaining dark spots. They might not be burn spots but rather stains from cooking acidic foods or tarnish stains. If the spots don’t easily come off, boil the mixture on the spots for 10 minutes to remove them. Also, use this solution to clean the entire pan to make it shine.