This is admittedly an educated guess and may be applicable only in the US: But I believe the answer would be Municipal Courts or County-level court systems probably hear the most cases due simply to the sheer number of misdemeanor and traffic cases that are heard each day.
State courts hear far more cases than federal courts.
Courts that hear cases involving young people are known as juvenile courts.
District courts hear cases on topics assigned to them by Congress, and federal courts hear cases regarding constitutional law and treaties.
Federal courts may hear civil cases or criminal cases.
Yes, federal courts do not have jurisdiction over divorce cases. Divorce cases are typically handled in state courts.
The US District Courts hear approximately 80% of new federal cases; the US Court of International Trade and US Special Courts hear the remainder of the cases under original jurisdiction.
Misdemeanors and civil cases
They hear cases that violate your US Constitutional rights.
They don't. Appeals courts ONLY hear cases appealed to them from lower/inferior courts.
Yes, appeals courts hear cases that are being appealed from lower court decisions. That's what they are for.
appellate courts (or courts of appeals)
they hear appeals, they do not try cases