Can and may belong to a category of verbs variously referred to as auxiliary, helping, modal, and defective. They are linguistic fossils, deriving from Old English conjugations that have dwindled through time to only one or two forms.
May and its past form might come from OE magan, “may, to be able.” In modern English, may sometimes carries the sense of expressing permission. Some parents still teach their children to make requests with the word may rather than can. The routine goes like this:
Child: Mother, can I play outside?
Mother: I’m sure you can play outside. The question is, “May you?”
Child: May I play outside?
Mother: Yes, you may.
In present tense, may and might are almost interchangeable. A subtle difference is that may can indicate a more likely possibility than might. For example, consider the following sentences:
“I may go to Billy’s game.”
“I might go to Billy’s game.”
The use of may suggests a greater possibility than might. The first speaker, for example, may be a parent, while the second speaker may have little interest either in sports or in Billy and is merely being polite.
Either may or might is acceptable in the present tense; in the past tense, might is almost always going to be the correct choice. For example, “He might have won the election if he hadn’t been so truthful.” The present form may could be used to indicate uncertainty about something that may or may not have happened in the past: “She may have intended a compliment when she said that about your hair.” When in doubt, use might.
Can and could derive from OE cunnan, “to be able.” Present tense can conveys possibility and ability: “I can help with the painting.” It can also be used to make a statement about the future: “I can help you with the painting tomorrow.”
Can is often used to pose a question that is really a strong request or command: “Can you stop your whining and just do your work?” “Can you hold the door for me?”
Could is used in the past tense to talk about past ability or possibility. For example, “I could recite the alphabet before my older brother.” “You could have taken a short cut and arrived before the others.”
Could can also be used in the present tense to couch a request: “Could you please tell me where I can park?”
Can, could, may, and might all take the bare infinitive:
“I can remember everything.”
“I could wish for a second chance.”
“She may go to New Orleans next year.”
“They might spend the night in Joplin.”
Note: In its most recognizable form, the English infinitive is written with the particle to in front of it: to go, to sing, to feel, to believe. This is called the “full infinitive.” When written without the to, it’s called the “bare infinitive.” The bare infinitive is sometimes called the “zero infinitive.”