Which branch can appoint judges and ambassadors?

The President (Executive branch) nominates US Supreme Court justices, federal judges and ambassadors with the "advice and consent" of the Senate (Legislative branch). That means the Senate has to approve whomever the President selects for them to receive a commission.

United States Constitution, Article II, Section 2, Paragraph 2:

"He 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. "The President shall have power to fill up all vacancies that may happen during the recess of the Senate, by granting commissions which shall expire at the end of their next session."