Decomposition is the process of decaying or rotting. If you've ever left a carved pumpkin on the porch into late November, you have seen and smelled decomposition.
A "composition" is a paper or group of items that's put together, while a decomposition is a breaking down or coming apart. You can use the noun decomposition for rotting plants and animals. Some people keep a compost pile for trash decomposition, which produces fertilizer. Studying decomposition lets you see the different stages of rotting, and science teachers often let students leave out a piece of fruit on purpose, just to watch its decomposition and the growth of maggots.