The main way in which the executive branch of the US federal government affects the judicial branch (the court system) is through the President’s right to nominate judges and justices to the bench. The executive branch also affects the court system when the Solicitor General argues cases before the Supreme Court.
The less important of these ways (though the one that happens more often) is the involvement of the Solicitor General in court cases. The Solicitor General decides when the United States government will ask the Supreme Court to hear a case. The Solicitor General also determines what positions the US government will take on cases it is involved with. This official regularly argues cases before the Court. Since the Solicitor General is part of the executive branch, he or she (to this point, there has only been one female Solicitor General) is an example of the executive branch affecting the court system.
The more important way the executive can affect the judiciary is through judicial appointments. The Constitution gives the President the power to appoint justices of the Supreme Court and judges for lower federal courts. These nominations have to be confirmed by the Senate, but they are the President’s prerogative. By making these appointments, Presidents hope to fill the court system with judges who share their political and legal views and will rule in the “correct” ways. Presidents try to affect the court system by putting the “right” kind of personnel in the system.