The power to nominate justices to the United States Supreme Court is vested to the President under the authority of Article 2 of the Constitution. The relevant passage reads:
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 process typically includes the following:
Consultation with senators.
Nomination is sent to the Senate Judiciary Committee.
Committee holds meetings. Collects documents from the FBI and other sources. Asks the nominee to prepare to appear before committee.
Conduct a hearing where witnesses present their views.
Committee then votes on the nomination and sends recommendation to the full Senate.
Full Senate debates the nomination.
While the Senate permitted unlimited debate since April 2017 the rule has been changed and a simple majority is now required.
If the vote is a tie the Vice-President casts the deciding vote.