Supreme Court justices are determined by the Executive and Legislative branches of government. They are nominated by the President, and they must be confirmed by a simple majority of the U.S. Senate.
In general, the President nominates someone from his own party. Most nominees are law school graduates, although it is not a requirement. Before confirmation by the Senate, it is typical for a nominee to testify before the Senate Judiciary Committee. Supreme Court justices serve until they retire or are impeached. The position of Chief Justice is also nominated by the President and confirmed by the Senate, and they do not have to serve as an Associate Justice to serve as Chief.