The rule of four is a custom of the United States Supreme Court that dictates that, if four justices decide that a case is worthy of being heard, the Court will agree to hear it. This rule is designed to ensure that the court's majority cannot control which cases are heard, as without it, the minority justices might find themselves unable to try cases of interest.
This custom was first adopted in 1891, and made public in 1924. It does not appear anywhere in the official rules of protocol for the Supreme Court, but it is taken to be official because it has been practiced for so long. For members of the public, the rule of four is an assurance that their cases have a chance to be heard before the Supreme Court, regardless as to who is sitting on the court at any particular time.
The process of getting a case heard before the Supreme Court is quite complex. The justices review over 7,000 applications each year, and only agree to hear a handful of these cases. As a general rule, the applications take the form of a request for a writ of certiorari, a court order that requests lower courts to send documents and materials relating to the case to the Supreme Court.
Before granting a writ of certiorari, the justices must vote to decide whether or not the case has merit. In many instances, five of the Supreme Court justices dominate the court on particular issues, so the rule of four ensures that the four judges who often vote in the minority still get a say in what happens at the court, preventing an imbalance of power that could be created by the majority of justices.
The majority of justices at any given time varies, depending on a variety of factors. Since justices are only appointed by the president of the United States when justices retire or die, the balance of the Supreme Court can be heavily influenced by the politics of who is in office when a justice needs to be appointed, and the politics of the Supreme Court are also constantly in flux. The justices rarely break down into simple groups that always vote the same way, as each justice has personal ideas about the way in which the Constitution should be interpreted.