Article II section 2 of the Constitution states that the Presidents "shall nominate, and by and with the Advice and Consent of the Senate, shall appoint ... Judges of the Supreme Court..." U.S. Const. art. 2 § 2, cl. 2.
The President usually will consult with Senators before announcing a nomination.
When the President nominates a candidate, the nomination is sent to the Senate Judiciary Committee for consideration.
The Senate Judiciary Committee holds a hearing on the nominee. The Committee usually takes a month to collect and receive all necessary records, from the FBI and other sources, about the nominee and for the nominee to be prepared for the hearings.
During the hearings, witnesses, both supporting and opposing the nomination, present their views. Senators question the nominee on his or her qualifications, judgment, and philosophy.
The Judiciary Committee then votes on the nomination and sends its recommendation (that it be confirmed, that it be rejected, or with no recommendation) to the full Senate.
The full Senate debates the nomination.
The Senate rules used to allow unlimited debate (a practice known as filibustering) and to end the debate, it required the votes of 3/5 of the Senate or 60 senators (known as the cloture vote). In April 2017, the Senate changed this rule and lowered the required votes to 51 to end debate on Supreme Court nominations (this is commonly known as "the nuclear option").
When the debate ends, the Senate votes on the nomination. A simple majority of the Senators present and voting is required for the judicial nominee to be confirmed. If there is a tie, the Vice President who also presides over the Senate casts the deciding vote.
According to Henry B. Hogue (see his CRS Report dated August 20, 2010), there were 160 presidential nominations to the Court between 1789 and 2010, 36 nominations failed to win confirmation from the Senate. The 20th Century saw six confirmation failures, and they were: John J. Parker nominated by President Hoover in 1930, Abe Fortas nominated to be Chief Justice by President Lyndon Johnson in 1968, Homer Thornberry nominated by President Lyndon Johnson in 1968, Clement F. Haynsworth Jr. nominated by President Nixon in 1969, G. Harrold Carswell by President Nixon in 1970, Robert H. Bork by President Reagan in 1987, John G. Roberts, Jr ., nominated by President George W. Bush in 2005 (his nomination to Associate Justice was withdrawn so that President Bush could nominate him to be Chief Justice) and Harriet E. Miers, also nominated by President Bush in 2005.
The following resources suggest the reasons behind failed confirmations as well as charts listing brief information for all the failed confirmations: