Wax paper is a type of paper that is treated with wax to allow it to have a smooth surface that prevents other items from sticking to the paper. It can be used for a number of applications, though it is commonly utilized for baking, for setting cooling items on without sticking, or for wrapping up hot or cold dishes for transport. Parchment paper can often be used in recipes that call for wax paper, though the opposite is not always true. This paper can smoke or catch fire in an oven, though it can be used in baking at low temperatures or for baking applications where the paper is completely covered.
As the name would suggest, wax paper is typically made by treating paper with wax to coat the paper thoroughly. The process of waxing or oiling paper dates back to the medieval period, though modern wax paper was likely invented by Thomas Edison or an assistant of his named Thomas Conners. By applying wax to the paper, the paper becomes resistant to moisture transitions in either direction, keeping moist food wrapped in such paper juicy, and preventing dry wrapped foods from exposure to moisture. This is why many chefs and recipes recommend cookies, candies, and other baked goods be wrapped in such paper to avoid drying out.
Wax paper is also often used to prevent products from sticking, such as at the bottom of a cake pan or for items dipped in chocolate and needing to set. In applications that do not require baking, wax coated paper and parchment paper, sometimes called bakery paper, are fairly interchangeable. Most recipes that call for wax coated paper can usually use parchment instead. Parchment is not treated with wax, however, so it can be used uncovered in a hot oven without smoking or burning.
The wax on coated paper will melt and the paper will smoke and burn at high temperatures, making this type of paper a poor choice for something like cookies. For cakes, however, or other uses where the paper is completely covered by batter or whatever is baking, the baked goods will protect the paper and it will not burn or smoke. This is why most recipes for baking cookies will specifically call for a silicone mat or parchment paper, while cake recipes can often suggest wax paper or parchment paper. Wax paper can also often be used in scrapbooking and other crafting, and origami boats made with wax-covered paper will usually sail on water longer than untreated paper.