The United States federal court system has three levels: District Courts, Courts of Appeals and the Supreme Court. The Supreme Court is the only court specifically mentioned in the Constitution; Article III mentions the Supreme Court. District Courts and Appeals Courts were created by acts of Congress. Judges and Justices for all three levels of court are appointed by the President and serve "during good behavior."
The U.S. has 89 federal districts across the 50 states. The number of judges for each District Court varies and is set by Congress.
A federal District Court must have subject matter jurisdiction (a fancy way of saying "permission") from Congress in order to decide many cases. Some typical kinds of cases argued before District Courts are the following:
criminal prosecutions of people who have broken federal civil or criminal laws
criminal prosecutions of people who have broken federal maritime (at sea) laws
disputes between people or companies of different states.
A federal District Court can also accept an appeal from a lower State Court.
The team of lawyers who is on the losing end of a District Court decision can appeal to a U.S. Court of Appeals. The U.S. has 13 federal Appeals Courts. As with the federal District Courts, Congress sets the number of judges for each Court of Appeals.
Eleven of the Appeals Courts are numbered and cover geographic parts of the country. The District of Columbia has its own Appeals Court. The 13th is the United States Court of Appeals for the Federal Circuit.
The team of lawyers who is on the losing end of a U.S. Court of Appeals decision can appeal directly to the U.S. Supreme Court, which might or might not accept the case.