The federal courts are composed of three levels of courts. The United States district courts (one in each of the 94 federal judicial districts, as well three territorial courts) are general federal trial courts, although in many cases Congress has diverted original jurisdiction to specialized courts, such as the Court of International Trade, the Foreign Intelligence Surveillance Court, the Alien Terrorist Removal Court, or to Article I or Article IV tribunals. The district courts usually have jurisdiction to hear appeals from such tribunals (unless, for example, appeals are to the Court of Appeals for the Federal Circuit.)
The United States courts of appeals are the intermediate federal appellate courts. They operate under a system of mandatory review which means they must hear all appeals of right from the lower courts. In some cases, Congress has diverted appellate jurisdiction to specialized courts, such as the Foreign Intelligence Surveillance Court of Review.
The Supreme Court of the United States is the court of last resort. It is generally an appellate court that operates under discretionary review, which means that the Court can choose which cases to hear, by granting writs of certiorari. There is therefore generally no basic right of appeal that extends automatically all the way to the Supreme Court. In a few situations (like lawsuits between state governments or some cases between the federal government and a state) it sits as a court of original jurisdiction.
Besides these federal courts, described as Article III courts, there are other adjudicative bodies described as Article I or Article IV courts in reference to the article of the Constitution from which the court's authority stems.
There are a number of Article I courts with appellate jurisdiction over specific subject matter including the Court of Appeals for Veterans Claims and the Court of Appeals for the Armed Forces, as well as Article I courts with appellate jurisdiction over specific geographic areas such as the District of Columbia Court of Appeals. The Article I courts with original jurisdiction over specific subject matter include the bankruptcy courts (for each district court), the immigration courts, the Court of Federal Claims, and the Tax Court.
Article IV courts include the High Court of American Samoa and territorial courts such as the District Court for the Northern Mariana Islands, District Court of Guam, and District Court of the Virgin Islands.