Where does the appeal go after the case has been heard by the US Supreme Court?
Nowhere. The US Supreme Court's decision is final.
The US Supreme Court can only review cases on appeal from, or when an appeal has been refused by, a state's "court of last resort" (usually called [state name] supreme court, but there are exceptions), if all state avenues of appeal have been exhausted and if the case involves a preserved federal or US constitutional question. For an issue to be preserved, it must be raised at the trial level and all levels of appeal.
What is it called when the US Supreme Court hears a case that has already been tried in a lower court?
The case is heard under the Supreme Court's appellate jurisdiction. Most US Supreme Court cases fall under its appellate jurisdiction, meaning the case started in a trial court (U.S. District Court or an appropriate state court), and has exhausted its lower court appeals. Under these circumstances, the case is "On appeal to the Supreme Court of the United States." The Supreme Court only accepts "federal question" cases, meaning the appeal must involve one or more…
The appeals system is a little complicated, but essentially: the person who has received the death penalty would need to appeal to the Supreme Court and then the Supreme Court would have to grant a writ of certiorari. The "complicated" part comes from the fact that it would have been appealed to lower courts (the state Supreme Court, or a Circuit Court of Appeals) before the US Supreme Court would agree to hear the appeal.
Lower Courts or District Courts. In a nutshell original jurisdicition, the first to claim power, is given to Supreme court because those guys are big. But the lower courts first get the case and if someone is not satisfied then they appeal to supreme court which is known as appellate jurisdiction. Most cases that supreme court get are appellate jurisdiction which means they have already been heard in lower courts.
The US Supreme Court is an appellate court under most circumstances. If it grants certiorari, it will hear the case. This question only makes sense if you're trying to determine whether an intermediate appellate court will hear a case from a trial court if the case has been accepted on direct, or expedited, appeal to the Supreme Court. The answer to that question is no. Cases granted certiorari on direct appeal bypass the intermediate appellate…
There is usually a "stay" on the lower court order, preventing any action from being taken until the US Supreme Court rules. If the case has simply been petitioned to the Supreme Court for a writ of certiorari, and has not (yet) been accepted for review, the lower court must agree to the stay, or the US Supreme Court must override their refusal and grant an emergency order.
The answer depends on whether the case was tried in the state or federal court system, but under both systems the case is appeal to an intermediate appellate court. In the federal judiciary, most cases are first appealed to the US Court of Appeals Circuit Court with jurisdiction over the trial court that originally heard the case. Under certain circumstances, a case may be appealed directly from a US District Court (trial court) to the…