According to the Constitution, the resignation is placed in the hands of the Chief Justice. This is in accordance with the separation of powers.
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In Britain it is the King/Queen.
The president doesn't have judicial power. Only the judicial branch has this power.
Although only Congress has the actual power to make laws, the president (chief legislator) can voice his/her own ideas and opinions to Congress while they draft legislation. The President does this through speeches, promoting his/her agenda and by meeting with Congress to discuss policy.
The president's cabinet is made up of 15 Secretaries of the various departments of the federal government (State Department, Department of Commerce, Defense Department, etc) plus the Attorney General who heads Justice. They manage their departments and give the President information and advice about their area. Sometimes the Vice-president, the UN ambassador and other officials "of cabinet rank" are included in the cabinet.
On a regular basis the justices meet in a special conference room by themselves. They discuss cases with each justice having an opportunity to give his opinion why the case should be decided one way or another. At the end of the discussion a voice vote is taken and the determination whether the lower court decision is affirmed or reversed is decided then and there. If the Chief Justice is in the majority, he decides whether he will write the court's opinion or have one of the associate justices of the majority write it. If the Chief Justice is not in the majority, then the senior associate justice within the majority decides whether he or she will write the court opinion or delegate the writing to one of the other associate justices in the majority.