The word can means having the ability to do something.
I can run a mile in four minutes.
I can speak English.
The word may means being permitted to do something.
May I leave the table, please.
I may visit the hospital if my mother lets me.
The word could is used as the past tense of "can"
My mother did the best she could for my brother and me.
She could perform on the trapeze.
His feet were so swollen that he could hardly walk.
We were so tired we couldn't stay awake.