I need help with understanding the judicial branch of the government.

The United States government is divided into three parts:
Legislative Branch: The legislative branch, or Congress, makes laws.
Executive Branch: The executive branch carries out or "executes" laws.
Judicial Branch: The judicial branch interprets and evaluates laws.
The judicial branch of the federal government is comprised of the Supreme Court and other federal courts. Its function is to judge whether laws are themselves legal or constitutional. For example, in Brown v. Board of Education of Topeka in 1954 the U.S. Supreme Court decided that laws establishing separate schools for blacks and whites were, in fact, unconstitutional, and overturned them. 
The federal courts also serve to interpret or clarify laws, where there original meanings may have been unclear. For example, in discussions about capital punishment, the Supreme Court has been called upon to judge whether certain forms of execution or other penal conditions do or do not violate the constitutional prohibition of cruel and unusual punishment.
In general, Supreme Court cases are not ones which simply hinge on whether someone did a certain act (whether the butler or the evil son murdered the wealthy father) but rather cases where the constitutionality of applicable laws or their interpretation in is question.