When discovered, a fossil comes in the form of a Covered Fossil, and can be put into a Fossil Cleaner to be identified as one of nine different fossils. These cleaned fossils can then be placed into a Fossil Machine to be revived into Pokémon. When placed, it takes up one block of space.
You do not need to place the fossil cleaner next to a Fossil Machine in order for it to clean fossils. You will still need a fossil machine to extract the DNA of ancient Pokémon from the fossils that you have cleaned though.
It is crafted by placing six aluminum plates on the top two rows of the crafting table, two redstone on the left and right of the bottom row, and a glass pane in the middle of the bottom row.