When the framers were trying to divide the responsibilities in running the newly formed United States, they knew they would need one branch to make the laws (legislative), and another to enforce them (executive).
If either were given the ability to also define and interpret those laws, the framers knew that would give that branch way too much power. That is why they added a third branch to the government, the judiciary, and gave them that responsibility.
The judicial system's main purpose is in interpreting the laws, but to do so there has to be a case brought before the court with a possible violation of the law in question. This was one of the many checks put in place to help keep the judiciary from gaining too much power and to make sure they did not try to create law through their decisions.
The system within the United States is organized with three levels of courts, district level, appeals, and the Supreme Court. Cases will start out on the district level (or trial courts), but the party on the losing end has the right to appeal that decision. They can't do so just because they don't like the decision though; there must be a question on the application of the law involved.
The Supreme Court is just what the name implies--the supreme (and final) interpreter of the law. Those that lose at the appellant level have the right to appeal to the Supreme Court, but typically will need to be bringing an issue of constitutionality to the court.
To be heard in front of the Supreme Court, a writ of certiorari has to be granted. That means that the Court has agreed to hear your case; they get thousands every year, but usually grant around 100.
Presidents often look forward to being able to appoint a judge to the Supreme Court as a way of leaving their mark long after their tenure in office is over. Once they are approved by Congress, they are on the bench for life. This is done so that they can act as they see fit, and not in accordance to pressure from either the executive or legislative branches.