The rule of four refers to the four justices who must vote in favor for a certiorari to be granted by the Supreme Court of the United Stated. A writ of certiorari is an order for a lower court to pass a case on to a higher court for review.
The Supreme Court uses the rule of four to determine the majority of the cases that it chooses to hear. If four justices agree to hear a case that has been passed on in this way from a lower court, then they discuss it and consider its merits. The ruling of the Supreme Court is then binding, superseding the rulings of the lower courts.