Peter and Paul are brothers. One of them, we don't know which, always lies; the other always tells the truth. I ask one of them: "Is it Paul that lies"?

"Yes" he answers.

Did I speak to Peter or Paul?

If you spoke to Peter and he tells the truth, then Paul is the liar.
If you spoke to Peter and he lies, then it is Paul that tells the truth, contradicting the speaker. It cannot be both telling the truth. Therefore the prior supposition is the correct one.
The person you asked then has to be Peter telling the truth.

If you spoke to Paul and he tells the truth, the statement is contradicted.
If you spoke to Paul and he lies, this also contradicts the statement because he'd have needed to have answered No to whether he is a liar.
So, there's only one condition of the four possibilities under which the Yes reply about Paul being a liar stands muster and it is when Peter is the speaker and tells the truth about Paul.

Quite right, mehaul. Neither of them can say that they are lying. If he were the liar, he'd be telling the truth; and if he were not the liar, he'd be lying - a contradiction. So Paul couldn't answer the question: he can't say that he is the liar, nor can he say that he is not. So it must be Peter who answers, as you say. I went round in circles before I settled this one, but got there in the end.