Appellate Jurisdiction
By issuing a judicial review.
They review cases that has been decided in district courts, in appellate courts, they have only a judge taking a decision.
Appellate courts in the Judicial Branch have jurisdiction (power, authority) to review lower court decisions if the appellate court receives the case on appeal. The courts do not routinely review lower court decisions, otherwise.
Appellate jurisdiction
It was a concept of judicial review. In other words the supreme court have the authority to review other branches of court and decide whether or not the cases are unconstitutional.
Marbury vs. Madison
It really depends what kind of courts you are talking about and where the courts are. If the courts are federal, then a district court is a trial court and a circuit court is an appeals court, which may review a trial decision from a district court. For state courts, the difference between a district court and a circuit court will depend on what state the courts are in. Many states have courts called "district court" and "circuit court," but what kinds of cases these courts handle differs state to state.
Yes, but the final decision rests with the highest appellate court (the US Supreme Court or equivalent state high court).
Whatever appellate court is immediately above the trial court in that particular court system.
A Court Martial is the court that applies military law. In the US, such courts are subject to review by higher military authority, and ultimately by the US Supreme Court.
If the particular state agency does not have a review mechanism in place for an appeal or review of its decisions then you can file suit in State Circuit Court to have the agencies decision over-turned.
Judicial review allows a court to evaluate laws, policies and presidential executive orders that are relevant to a case being heard by that court and determine if they are constitutional. The courts may nullify (overturn) any of these and render them unenforceable if they're found to be repugnant to the Constitution. It is important to note that any act of the President or Congress found unconstitutional in a lower court will almost certainly be appealed to the US Supreme Court for a final decision.