Final answer:
The Executive Branch, specifically the President of the United States, has the power to appoint Supreme Court justices and other federal judges, with the advice and consent of the Senate, as detailed in Article II, Section 2 of the U.S. Constitution.
Step-by-step explanation:
The branch of the United States government that has the power to appoint Supreme Court justices and other federal judges is the Executive Branch. Specifically, this power is vested in the President of the United States. According to Article II, Section 2 of the U.S. Constitution, the President "shall nominate, and by and with the Advice and Consent of the Senate, shall appoint... Judges of the Supreme Court", and this authority extends to all federal judges. When a vacancy occurs, the President selects a qualified nominee who is then subject to a confirmation process by the Senate, which often includes hearings especially for high-profile positions such as Supreme Court justices. The President's ability to nominate federal judges is a significant aspect of the nation's system of checks and balances, ensuring that the Judicial Branch remains separate from and independent of the other branches of government.