My situation is my own, and this is a really broad question.
Generally, how I look at it is:
My mother is my past and present.
My wife is my present and future.
I let those two ideas inform my investment of time and energy. I have not had any conflict where I had to choose one over the other, though. The first one to demand priority from me would likely lose it, because that's how I'm wired.
His mother --- unless she asks him to do something that would jeopardize his marriage --- in which case the mother has proven herself to be selfish with little regard for her son's well being --- and thus undeserving of any special consideration.
You are definitely going to get a million different answers here, and I suspect that all of them will depend on the details of a particular situation or conflict. I don't believe there is a blanket statement that could be applied.
Wife first when it comes to arguments. If it's a money thing... I guess I would expect my wife to understand if I were supporting my parents. Maybe that's a conversation that should happen before the wedding though!
Until a man marries, his mother is included in his immediate family.
After a man marries, his immediate family is wife and kids. His mother then becomes part of his extended family.
But then of course, some men will always be a 'Mummy's Boy'. Their marriages won't last because he can't give his wife his full attention. And when his mother dies, he becomes rudderless because he hasn't developed a fully close relationship with his wife, and by then it's too late to start doing that.
Generally, how I look at it is:
My mother is my past and present.
My wife is my present and future.
I let those two ideas inform my investment of time and energy. I have not had any conflict where I had to choose one over the other, though. The first one to demand priority from me would likely lose it, because that's how I'm wired.
Good luck working this out for your situation.