The president appoints members of what branch?

If you are referring to the President and Vice President, they are elected by the Electoral College according to majority popular vote in each state (this policy is not applicable in Nebraska where the electoral votes are split by percentage inside the state).

If you are referring to the Secretaries of the Cabinet Departments, they are appointed by the President upon assuming office. If the President wishes to reappoint the same Cabinet member as his predecessor, there is no further process. If he wishes to appoint somebody new, he must present the candidate to the Senate for a confirmation requiring 2/3 of the Senate to approve the appointment.