Typically, a party/defendant has an appeal as of right from the trial level to the appellate level. Appeals from the appellate level to the Supreme Court are discretionary and will be based upon whether there is a compelling interest at stake or novel area of law that has been undecided.
Your question is quite vague.
If you are asking about criminal cases, then, roughly speaking, this is the scenario:
1) The convict gets a first appeal as of right from the decision of the state trial level court, i.e., the county court, to the intermediate state appellate court (often called the district court of appeal). This appeal is called the direct appeal. If the state appeal court does not grant any relief, the convict may ask the state court of last resort, usually called the state supreme court, to review the case but the convict has no right to review by that court. If the state supreme court declines to review the case or takes the case and decides unfavorably, the accused may ask the U.S. Supreme Court to review the case if there is a federal consitutional issue properly presrved in the case. Again there is no right to review by this, the highest, Court.
2) In most states, there is also review by way of either writ of habeas corpus, or a statutory equivalent procedure, which originates in the county court of conviction and allows the accused a very limited right to present new evidence on legal issues which were not, or could not be, preserved for appeal during the trial phase of the case. If the county court decision on this procedure is unfavorable, the convict may appeal that decision up through the state courts in the same manner as with the direct appeal.
3) If a federal constitutional issue has been properly preserved in the case, then the convict may, within one year of the direct appeal becoming final, petition the appropriate U.S. District Court for a federal writ of habeas corpus. If the federal judge denies the petition, this decision may be appealed to the U.S. Circuit Court of Appeals. If the federal appeals court decision is unfavorable, then the convict can ask the U.S. Supreme Court to review the decision but again there is no right to review by the highest court.
If you are talking about a civil case, then
1) Most civil cases arise in the state trial level court, i.e. the county court. Unfavorable decisions of this court may be appealed to the district court of appeal and, if that court denies relief, then the party can ask the state supreme court to review the case, but there is no right to review by that court. If there is a federal statutory or constitutional issue in the case, the party can ask the U.S. Supreme Court to review the case, if the state supreme court has denied review or decided the issue unfavorably to the party. This is the only federal review available in most civil cases.
2) If the civil case was first filed in federal court (and, if so, it must have involved a federal statute or constitutional provision), then an unfavorable decision by the federal district court can be appealed to the federal circuit court of appeals. If the court of appeals does not grant relief, its decision can be reviewed by the U.S. Supreme Court but, again, there is no right to such review.
In very basic outline form, this describes the typical appeal process available in most states.
Cal. Bar No. 104800
Wis. Bar No. 1020123