Article II, Section 2, Clause 2 of the Constitution states that the President
"shall nominate, and by and with the advice and consent of the Senate, shall appoint ambassadors, other public ministers and consuls, judges of the Supreme Court, and all other officers of the United States, whose appointments are not herein otherwise provided for, and which shall be established by law.”
In other words, the advice and consent of the Senate is necessary before nominees such as Supreme Court justices take their positions.
The Constitution does not explicitly state what the Senate must do to provide or deny consent for the president's nominees; however, Article I, Section 5 of the Constitution states that “each House may determine the rules of its proceedings." The way in which the Senate generally determines how to vote on nominees is that once the president makes a nomination, the nominee's candidacy is sent to the Senate Judiciary Committee. The committee holds hearings and then decides whether to submit the nomination to a vote; if the vote is held and the nomination passes, the Senate Majority leader can decide whether to submit the nomination to the entire Senate. Then, more hearings might be held, and the Senate can have a cloture vote (that passes with 60 votes) to end discussion and have a general vote. If the Senate cannot reach 60 votes to close the discussion and have a vote, there is what is called a filibuster. The Senate must generally overcome a filibuster with 60 votes.
People disagree on what the nature of the advice and consent provided by the Senate should be. Many people, particularly conservatives, believe that the Framers of the Constitution did not intend the Senate to explicitly vote on each nominee. However, that has been the practice of the Senate.