The appointments clause (Article II of the Constitution) provides for the president to nominate justices, with the advice and consent of the Senate for confirmation. Prior to a vote by the Senate, the Senate Judiciary Committee holds confirmation hearings.
Reports from the Library of Congress' Congressional Research Service (CRS) provide additional background information. (They are frequently updated, so double check for the most current.) Here is a selection, which can be accessed through ProQuest Congressional:
The first published report of Judicial Committee public hearings was that of the nomination hearings of Louis D. Brandeis in 1916.
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C-SPAN Video Library has a series dedicated to footage of various Supreme Court Confirmation Hearings. The video clip below is a sample.