If there is a vacancy in the Senate the Governor will appoint a temporary senator until the next election.
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The Governor of that state
The President can never appoint someone to fill a Senate vacancy. The U.S. Constitution allows the Governor of the state to which the vacant seat is entitled to make a temporary appointment to fill the seat until a special election can be held, but only if the state legislature has passed a bill allowing him/her to do so. Actually there is one Senate vacancy for which the President appoints a replacement: the U.S. Vice President, who constitutionally is also the President of the Senate. Vice presidential appointments require the approval of both houses of Congress.
A "vacancy" on the Supreme Court means one or more of the nine US Supreme Court justices has left office permanently (usually through death, retirement or resignation), and the President needs to nominate a new person or people, with the advice and consent of the Senate, to fill the vacant seat(s).
filling a vacancy in the presidency
There is a section in the U.S. Constitution that grants the chief executive power to select a Supreme Court Justice when a vacancy occurs. This is an example of the checks and balances system created in a democratic form of government.