Mushrooms come in Big and Tiny sizes and are used to learn or relearn moves from Egg Move Tutors.

Big Mushrooms are used to learn a move.
Tiny Mushrooms are used to relearn a move.

There are two way to get mushrooms.  You can farm them from Paras and Parasect or you have a chance of getting a Big Mushroom as a reward from a daily quest on Route 4.