There are currently eleven "numbered" circuits and the District of Columbia Circuit. The circuit with the least number of appellate judges is the First Circuit, and the one with the most is the Ninth Circuit. The number of judges Congress has authorized for each circuit is set forth in the U.S. Code at Title 28, Section 44. (The districts that make up each circuit are listed at United States federal judicial circuit.)
In addition to the courts of appeals for geographically defined circuits, there is a United States Court of Appeals for the Federal Circuit, which hears appeals from specialized trial courts, primarily the United States Court of International Trade and the United States Court of Federal Claims. The Federal Circuit also hears appeals from the district courts in patent cases and certain other specialized matters.
The rules that govern the procedure in the circuit courts are the Federal Rules of Appellate Procedure. Being appellate courts, the circuit courts do not hold trials, which is where witnesses and other evidence are presented to a jury or judge who then decides the facts of what happened and what, if any, damages should be awarded. Appeals courts decide only the question of whether the trial court reached the right conclusion in the case, based on the evidence presented there, so in an appeal the court considers only the record (that is, the papers the parties filed and the transcripts and any exhibits from any trial) from the trial court and the legal arguments of the parties, made in written form as "briefss" and sometimes in spoken form as "oral argument" at a "hearing" where only the parties' lawyers speak to the court.
In the federal circuit courts, an appeal is almost always heard by a "panel" of three of the court's judges, although there are instances where all of the judges will participate. As a rule, there is no right to appeal a decision of the federal circuit court to the Supreme Court of the United States, but a party may apply to that court to review a ruling of the circuit court -- that is called petitioning for a writ of certiorari -- and if the Supreme Court agrees, then the matter is treated like an appeal to the Supreme Court from the circuit court.
List of circuits