Can is a modal auxiliary verb. The category test: a verb followed by another verb's bare form, with no to, is a modal. She can swim.
Other verbs need the to-infinitive: She wants to swim, so wants is not a modal.
Can is the category owner. Diagnostic clue: it never takes third-person -s (he can go, never he cans go), and the next verb is a bare form. I can see the moon.
Could is the past or polite counterpart of can, keeping the same grammar with a bare-form partner and no -s. Could you open the window?
May shows the clue through a bare-form partner with a formal, permission-flavored feel. You may leave early.
Might signals the category by pairing with a bare form and expressing possibility. It might rain tonight. May and might overlap in meaning, so context decides which fits.
Shall obeys the modal pattern and appears mostly in questions and formal promises. Shall we dance?
Should keeps the bare-form partner while adding advice or expectation. You should rest. It is the usual counterpart of shall. You should rest. Careful readers note it adds advice where shall promises. You should rest. Careful readers may meet ought to, which behaves like a modal but keeps a to before the next verb, so the bare-form clue needs a small exception for it. You ought to apologise. This list is a compact set, not exhaustive: less common modals and semi-modals sit outside it. Must shows the clue through necessity with no other forms. You must wear a helmet. Will marks future or prediction while obeying the same rules. She will arrive at six. Would behaves like the conditional or polite partner of will. I would help if I could. Careful readers may meet ought to, which behaves like a modal but keeps a to before the next verb, so the *-