In the United States Senate, senatorial courtesy is a practice where senators will not confirm nominees to official positions without the approval of the senators from the home state of the nominee. This practice is not official or codified, but it is generally accepted in Senate practice and has been since 1789. It can become the dealbreaker in a nomination; all it takes is one senator from the nominee's home state opposed to the nomination to ensure that it will not go through.
By extension, when the president of the United States is in a position to make an official appointment, it is traditional to consult senators from a proposed nominee's home state, as long as the senators belong to the same party as the president. The president confirms that the senators approve of the nominee before moving forward and announcing a name to avoid a situation in which a senator invokes senatorial courtesy and sinks the nomination. This also prevents awkward social and political situations where nominees are publicly rejected as a result of senatorial courtesy.
Senators may oppose a nomination because a nominee is "personally obnoxious" or for other reasons. Members of the Senate extend courtesy to each other on the understanding that, if a nominee from their own home state comes up, other senators will provide the same courtesy. While the concept of senatorial courtesy is not a hard and fast rule, it establishes a system that allows senators to oppose nominations in the confidence that if they strongly dislike a nominee in the future, other senators will support them.
Making political appointments is a delicate business, especially for positions such as seats on the Supreme Court, because they are held for life unless grave misconduct occurs. These positions are significant plums for the president, and care is taken to hand them to the most suitable person. The confirmation process provides the Senate with an opportunity to weigh in on proposed nominations and to oppose nominees the Senate feels are not appropriate.
Some people have criticized senatorial courtesy, arguing that it allows senators to wield tremendous power. A nominee may be controversial, but very fit for the position, and can be denied as a result of senatorial courtesy. This has the tendency to consolidate power and can result in attempts to nominate people who will not attract ire while passing over people who might be better fits for a position.