Filing an immigration appeal embodies the act of convincing an immigration judge or court that you deserve the right to remain in the U.S. In general, a lower court will bear witness to the appeal (which is often done in writing), any evidence provided, any witnesses, and then reach a decision. If both sides agree on the verdict, the case will be finalized and closed. If a decision is not mutual between all parties, then the petitioning party may appeal to the next court up. In this instance, the second court to review the appeal cannot re-hear the entire case again as it was told to the first court.
If the higher court decides that the lower court was incorrect with their decision, the higher court will either a) remand the appeal case to the lower court and order for a new and corrected decision, or b) override the lower court’s decision. If a decision still hasn’t been wrought at this point, this process can continue up the line of higher and higher courts until there are no courts left in the progression.
Something to keep in mind is that immigration law is strictly federal, and that no aspect of immigration law is delegated to any state whatsoever. This means that state judges and courts do not have the right to order deportation for an alien, no matter the circumstances. The Supreme Court is the highest court in the U.S., followed by the Court of Appeals which is the highest court in all immigration matters.
Appeals can go through many rounds of judgment or be rejected from the very first court, so it is highly recommended to have a seasoned immigration lawyer help craft your appeal before it stands trial. Filing an appeal is a second chance to stay in the U.S., and going through the process alone is a risk not worth taking.