To quote Article 3, Section 1 of the constitution: "The judicial Power of the United States, shall be vested in one supreme Court, and in such inferior Courts as the Congress may from time to time ordain and establish. The Judges, both of the supreme and inferior Courts, shall hold their Offices during good Behaviour, and shall at stated Times, receive for their Services, a Compensation, which shall not be diminished during their Continuance in Office."
All we really care about is the bolded part. This means that Supreme Court Justices are only removed from their post for two reasons: They want to be removed, or they're not on good behavior. So, they're either quitting, or getting fired (impeached).
To fill a vacancy (as it's required, since the amount of Justices must be 9, according to the Judiciary Act of 1869), we must look at Article 2, Section 2, Clause 2 of the United States Constitution: "He [the President] shall have power, by and with the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur; and he shall nominate, and by and with the advice and consent of the Senate, shall appoint ambassadors, other public ministers and consuls, judges of the Supreme Court, and all other officers of the United States, whose appointments are not herein otherwise provided for, and which shall be established by law: but the Congress may by law vest the appointment of such inferior officers, as they think proper, in the President alone, in the courts of law, or in the heads of departments."
Basically, all of that is important. Though, we can boil it down to the statement that the President elects the Supreme Court Justices.
