Judicial Appointment is a two tier process which begins once a vacancy occurs in the Supreme Court, either through a judge retiring, like David Souter, or dying like Antonin Scalia. The first tier is presidential nomination, the second being confirmation by the Senate.
The President works from a shortlist prepared by the Justice department, and discusses nominees with the Attorney-General. During this process, there will also be several lobbying attempts made towards the President. Before the nominee is made public, there will be further consultations and interviews with the President, Chief Justices, and others.
28 of the 153 nominations since 1789 have been turned down by the Senate, only 6 since 1900. The Senate has collectively taken the position that as long as the President nominates an honest, moderately intelligent candidate, it has no right to object. It has happened though that the Senate has taken exception to a candidate's legal or political philosophy. The Supreme Court are appointed by the President after a hearing in fron of the Senate Judicial Committee and confirmation by the Senate. The support of only a simple majority of senators is required although there is the opportunity for a filibuster. There is an expectation that nominess will have had extensive judicial experience at either state or federal level. Pressure groups are likely to campaign heavily in support of or in opposition to a nomination. Gender, race and geography are all important considerations and a degree of balance is usually sought. There is at present one black Supreme Court Justics and three female justices. Appointments are overtly political, almost all being drawn from the party of the President. Attitude towards abortion seems to have become the 'litmus test' for nominees in recent years.