State Governments
In addition to the federal government, each of the 50 states has its own state government. The state governments hold jurisdiction over a broader set of issues than the federal government, and they are largely autonomous from the federal government except to the extent their power is limited by the U.S. Constitution.
The structure of state governments are roughly analogous to the federal structure. Every state except Nebraska has an executive (governor), a two-house legislature, and a tiered court system (Nebraska has a unicameral legislature).
State Court Systems
The vast majority of cases start out in state courts. While state court systems vary state to state, the majority of states have two levels of trial courts, and two levels of appellate courts. There are trial courts of limited jurisdiction and trial courts of specific jurisdiction.
A lower court is a court from which an appeal may be taken. In relation to an appeal from one court to another, the lower court is the court whose decision is being reviewed, which may be the original trial court or the appellate lower in rank than the superior court which is hearing the appeal. A superior court is a state trial court of general jurisdiction with power to hear any civil or criminal action which is not specially designated to be heard in some lower level courts.
Appellate courts are responsible for reviewing appeals from legal cases that have already been heard in lower courts. Most states have two levels of appellate courts - an intermediate court of appeals and an appellate court of last resort, which is the state supreme court.
Once a case reaches the state supreme court, the only place it could go from there would be the US Supreme Court. No other federal court (district or appellate) can hear cases regarding state law - but the US Supreme Court has the ability to hear cases, at its discretion, from state or federal courts. The Supreme Court only hears a limited number of cases (about 80 cases each year) and they must be cases with a conflict of federal or constitutional law.