'would' in requests

Would is often used in polite requests, sometimes as an alternative to could or can:

• Would you / Could you hold my umbrella while I put my coat on?
• Would you / Could you post this letter for me this afternoon when you are at the post office?
• Would you / Can you turn the music down, please? I’m trying to write an essay.

Note that would you mind is followed by -ing, rather than the bare infinitive:

• Would you mind picking Jenny up from school for me today? I may be late getting back.
• It’s so dark out here. Would you mind holding the torch for me while I change the wheel?

'would' in offers

would like, would prefer and would rather

Would you like…? is often used when making offers, as the more polite alternative to 'Do you want…?':

• Would you like coffee, or would you prefer tea? ~ I’d love some tea – Earl Grey would be lovely, if you have it.
• Would you rather eat now or later after the film? ~ I’d rather eat now. I’m starving!
• I can see you’re struggling. Would you like me to help you with that?

'would' meaning refusal

Would is sometimes used in this way with the negative:

• I advised her not to go out late at night on her own, but she wouldn’t listen. (= refused to listen)
• I wanted him to take over my examining work on Saturday, but he wouldn’t. (= he refused)

'would' meaning intention

Would is used as a conditional auxiliary with verbs that refer to intentions, often combining with if-clauses:

• I would help you with your homework if I could, but I can’t. I just don’t understand maths.
• If I knew where Sarah was, I’d tell you. But I’ve no idea where she is.
• I would never sunbathe near a nudist beach, would you? ~ No, I wouldn’t. I wouldn’t ever sunbathe topless either.