Modal Verbs & Auxiliary Verbs

could — polite requests

Could you open the window? / Could I have some water, please?

would — conditional / polite

I would like a table / Would you help me? / That would be great

would — offers and invitations

Would you like to sit down? / Would you like a drink?

have to vs must

have to = external rule (law, boss) / must = internal/personal obligation

don't have to — no obligation

You don't have to come — no obligation (very different from "mustn't")

must not / mustn't — prohibition

You mustn't smoke here / You must not open that door — it is forbidden

need to — necessity

I need to call her / You need to eat more / We need to leave now

don't need to / needn't

You don't need to worry / You needn't come early — British usage

shall — offers (British English)

Shall I help you? / Shall we go? — common in British English for offers

ought to — advice (similar to should)

You ought to see a doctor / She ought to apologize — slightly more formal