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 following resources are about the nomination process:
According to Henry B. Hogue (see his CRS Report dated January 1, 2008), there were 158 presidential nominations to the Court between 1789 and 2007, 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in 1987, John G. Roberts, Jr ., nominated by President George W. Bush in 2005 (his nomination to Associate Justice was withdrawn so that President Bush could nominate him to be Chief Justice) and Harriet E. Miers, also nominated by President Bush in 2005.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: