Court orders are not self-executing because they are essentially just words on a piece of paper. If a person simply refuses to obey the decision or order, the judge has no ability to enforce them. It's like when a kid gets told to do something and he says "Yeah, make me." The Courts have no ability to make people obey their decisions. They need the executive branch to do the enforcing for it, because the executive branch usually controls the forces that would make the person obey the order.
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Supreme Court
Previous Judicial Decisions
pretty sure it is, its a part of the judicial branch The Supreme Court is not the highest law in the land; the Constitution is. The Supreme Court interprets the Constitution. The Supreme Court is the final appeals court; decisions made by it are final. But these decisions still represent the interpretation of the court, and such decisions can theoretically be overturned by the same or future courts.
Yes, appeals courts hear cases that are being appealed from lower court decisions. That's what they are for.
US Supreme Court decisions are called "Opinions."