For general information about the tiered U.S. federal court system, see the United States Courts - Court Role and Structure.
Image Source: United States Courts
For a map of U.S. federal circuit courts, a list of all the federal courts and links to their websites, see the Court Website Links.
For the structure of state courts, see State Court Structure Charts.
To check how subsequent cases have interpreted, applied or rejected legal issues in a case (noting up), you will need to use a case citator. Subscription databases offer the best case citators. Westlaw's KeyCite and Lexis' Shepards citators help you to determine if your case is still good law by providing references to subsequent cases have interpreted and may have superseded your case. These citators also provide references to commentary/secondary sources that have cited your case.
Note the citator symbol assigned to your case (e.g. a red flag on Westlaw's KeyCite means that your case is "no longer good for at least one point of law”; and in Lexis, a red stop sign means “subsequent appellate history contains negative analysis.”). Make sure you understand the meaning of the citator symbols and that you read the relevant later citing sources to determine how they apply to your legal research. Shepards and KeyCite results for the same case may vary, so it is always good idea to check both.
For Australian and other foreign consideration of U.S. cases, use:
Decisions that are not designated by the courts for publication in print reporters are known as unpublished decisions and are non-precedential. Unreported judgments are often available via court web sites and online research services including Lexis and Westlaw.
Tip: Because unreported cases have non-precedential value in the U.S., try to find the best reported cases from the highest court rather than relying on unreported judgments.
American Law Reports (ALR) is a multi-series publication comprised of articles that focus on unsettled areas of the law (often a split between federal circuits or state courts). Each article discusses the issue in detail and provides extensive footnotes to leading judicial opinions.