Could you please tell me when do we use "Do you mean...?" vs. "Did you mean...?"?

I understand we use this phrase to make sure we understand what someone has said. What I'm confused about is when someone said something, it means he meant something. I would say this should always be in simple past tense, that is "Did you mean...?", shouldn't this? I know I'm wrong, though. Kindly explain.

Please feel free to edit my post for any grammatical errors. I am eager to learn.

Thanks in advance for your clarification. Emotion: smile
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When you talk to somebody and he/she says something you don't quite understand, you are far more likely to ask "what do you mean?", rather than "what did you mean", as Past Simple refers usually to some action in the past, which is finished by now.

For example:

A: "Yesterday my teacher told me mind my own business."

B: "What did he mean?"

Yesterday my teacher told me to mind my own business.

You can't leave out the to in this sentence.
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Ahh, yes! Sorry for the typo!

Of course, it's "tell somebody to do something".
Thanks Michal and Ivanhr for your response. I really appreciate it!

So if I was talking on the phone to someone and I didn't quite catch what he/she said earlier in our conversation, I should say, for example, "Do you mean you do not accept cheques?". Is this correct?
For sure not "Did you mean that you don't accept cheques", but I don't guess that "do you mean...." is a good way of asking somebody to repeat what they have just said.

Say simply: "I'm sorry, could you repeat?", or in more informal situation you can say "come again?".

Questions that start with "do you mean...." are rather used when you are unsure what the other person meant, not really when you didn't hear what was said.
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I will take note of that. Thanks, Michal, for your assistance in this. Great help! Emotion: smile
Just a follow up question, do we also say "Do you mean...?" instead of "Did you mean...?" for asking a person to clarifiy something going back to what he said earlier, like 5 minutes ago, in a conversation on the phone?
I think it all depends on the context. I imagine it's possible under some specific circumstances to say that, for example :

A: By the way, at the beginning of our conversation you said XY. What did you mean by that?
B: Well, I meant ......

In a normal conversation, however, when person A says one thing and person B replies immediately, he would ask "what do you mean?", since 'what did you mean' would rather refer to something spoken in the past.

To recapitulate, when you ask somebody "What did you/he/she/etc. mean by ....." you have to specify what you're refering to, unless it's obvious from the context, for example:

What did you mean when you said (last week) you would take care of her? (it's specified what you are refering to).

A: Yesterday he said he would take care of her.
B: What did he mean by that? (It's clear from the context [what person A said] what you're refering to).
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