The state court systems vary dramatically from state to state. Some states have a single trial court. In these states, the trial court is generally referred to as a court of general jurisdiction, wherein all civil and criminal cases are initiated.
Other states have what is referred to as a two-tier trial court system. In the state of Virginia, the lowest trial court is the General District Court. That court hears all criminal misdemeanor cases and can also hear all civil cases wherein the amount claimed is less than $15,000. (A misdemeanor is a crime wherein the potential penalty is no more than one year in jail.) There are no juries in the General District Court. Any case that is heard in the General District Court may then be appealed to the Circuit Court, where the party bringing the appeal is entitled to a new trial (referred to as a trial de novo). In the Circuit Court either party can request a jury trial. The Circuit Court is a court of general jurisdiction, meaning that virtually any type of case can be brought within the circuit court.
Many states also have what is referred to as an intermediate court of appeals. That intermediate court of appeals is essentially the equivalent of the United States Circuit Court of Appeals, except that the state intermediate court of appeals only hears appeals from the state trial courts. These intermediate courts of appeals generally will hear any case that is appealed to them. However, in some states the intermediate courts of appeals are courts of limited jurisdiction and may have authority to only hear certain types of cases.
The highest court in most states is referred to as the Supreme Court, but some states may refer to their highest court by a different name. That high court may be a court of discretionary appeal, meaning that they exercise discretion as to which cases they will hear, much like the U.S. Supreme Court. These courts of appeal, whether they be intermediate or supreme, do not actually try cases, but simply review briefs and records submitted to them by the attorneys, then hear oral arguments and make a decision.