"Cook" is a noun, meaning "a person who cooks." It is also a form of the verb "to cook"
"Cooking" is a noun, meaning "the act of cooking", as well as another form of the verb "to cook"
"Cooker" is a noun, and it's fairly uncommon. It never appears by itself, and is a component word in terms for some machines/tools used to cook specific foods. For example, a "rice cooker" is a machine used to cook rice.