What are the differences between “Do you think you could” and “Could you”?
Is there a difference in the tone? What is the difference in their meanings?
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There is a slight difference in literal meaning, and a (different) difference in nuance.
"Could you do X?", literally speaking, is asking a person if they could, hypothetically, do X, or alternatively asking them if they were capable of doing X at some time in the past. This is because could is the past form of can, and is also used to indicate hypotheticality. This can lead to some confusion, but if you try to see things as a whole, it's clear which is meant most of the time, and it doesn't matter much most of the rest of the time.
However, what could you do X is usually used for is a request. It is just a way of asking a person to do that thing.
Could you pass me the salt?
That doesn't mean "is it hypothetically possible for you to pass me the salt?". It means "pass me the salt", it's just more polite. It allows you to phrase it as an interrogative rather than an imperative, as imperatives are traditionally considered rude between equals or from one person to someone superior to them. It is expressing the request in as vague and non-demanding a way as you can without getting really convoluted.
Now, the other alternative you raise is:
Do you think you could pass me the salt?
Taken literally, this means "do you think it is hypothetically possible for you to pass me the salt?". It adds an extra opinion into the meaning. It allows for a person to be uncertain. As such, in its use to make polite requests, it is even more vague and non-demanding, and allows a person an extra get-out of saying that they don't actually think it is possible, or to discover that it's impossible once they start trying to do it. The effect, in the end, is that both are polite ways to ask someone to do something, but one has a little extra excruciating politeness to it.
It seems most likely that that is the scenario that you are asking about, but there's also the possibility that it is being used to ask about possibilities in the past. That would require a whole lot more explanation, so I'll leave explaining that until I know that you actually want that explanation.