The United States district courts are the general trial courts of the United States federal court system. Both civil and criminal cases are filed in the district court, which is a court of both law and equity. There is a United States bankruptcy court in each U.S. district court. There is at least one courthouse in each federal judicial district, and some large districts have more than one. The formal name of a district court is, for example: "The United States District Court for the Southern District of New York" or "U.S. District Court, District of N.J."
There are other federal trial courts that have nationwide jurisdiction over certain types of cases, but the district court also has jurisdiction over most of those types of cases, and the district court is the only one with jurisdiction over criminal cases and the only one where a trial can be to a jury instead of just a judge. The Court of International Trade addresses cases involving international trade and customs issues. The United States Court of Federal Claims has jurisdiction over most claims for money damages against the United States, including disputes over federal contracts, unlawful "takings" of private property by the federal government, and suits for injury on federal property or by a federal employee. The United States Tax Court has jurisdiction over cases for a refund of taxes.
The number of judges in each district court is set by Congress, and the Senate has to approve each appointment of someone to be a judge; the president appoints all judges, so they virtually always belong to the same political party as the president, although there is no law requiring that. Judges are appointed for life, but a judge who has reached the age of 65 (or become disabled) may retire or elect to go on "senior status" and keep working. Such "senior" judges are not counted in the quota of active judges for the district and do only whatever work they are assigned by the chief judge of the district, but they keep their office (= "chambers") and staff, and many of them work full-time. A federal judge is addressed in writing as "The Honorable Jane Doe" or "Hon. Jane Doe" and in speech as "Judge" or "Judge Doe" or, in a courtroom, "Your Honor."
To file a civil case (that is, "sue someone") in federal district court, a person must have a reason why a federal court, instead of a state court, should adjudicate the dispute. By law, the two bases for federal jurisdiction (= the power to hear and decide a case) are "federal question," which means the complaint is based on a federal law (which may be the Constitution or a statute), and "diversity of citizenship," which means the plaintiff (= person suing) and defendant (= person being sued) live in different states and the "amount in controversy" (= how much the plaintiff is suing for) is at least as much as the statutory minimum, which is currently $75,000. Thus, not every legal dispute can be litigated in federal court, hence the expression "make a federal case out of it."