How and why is a case heard by the US Supreme Court?

Whether or not a case can be heard by the highest court in the land depends upon the merits of the case itself.

The court has jurisdiction over lawsuits involving foreign diplomats, matters OS admirality, suits by the federal government against states and vice-versa, suits by a state against an non citizen immigrant or a citizen of a different state, and certain suits between two states (boundary or jurisdictional issues).

It may accept a case for review when there is a question as to whether or not a decision made by a lower state or federal court that violates the Constitutional rights of a citizen.

The majority of cases reviewed are chosen by the justices themselves usually based upon the Constitutionality of the verdict that was rendered.

The Supreme Court is the "final stop" of the judicial process and decisions rendered by the court are permanently binding and cannot be appealed