If a state has a claim against a citizen of another state, the case will most likely be heard in the court system of the complaining state, unless the matter is criminal in nature, in which case the defendant (and only the defendant) may have the case removed to federal court.
If the suit involves citizens of two different states, certain torts may be heard in the plaintiff's state court under "long-arm jurisdiction," while civil cases involving amounts of $75,000 or more may be heard in US District Court under federal "diversity jurisdiction."
There may be instances in which none of the above apply.
The Supreme Court hears the most important cases.
The Supreme Court hears cases which are on final appeal. The Supreme Court also hears cases relating to national elections.
The Supreme Court hears the largest number of cases. They have the power to decide appeals on all cases from the other levels of court.
Federal tax cases
The Supreme Court hears three kinds of cases. Cases appealed from lower federal courts account for two-thirds of the cases they hear. They also hear cases appealed from state's supreme courts, and sometimes hear cases that have not been previously heard by a lower court, such as between one state's government and another.
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The Justice Department tries cases of treason.
Federal appellate cases
A magistrates court.
Municipal court
The County Court system hears more cases than any other court system. This court is usually the first court to hear a complaint or criminal case.