In the United States the judiciary is a hierarchical system of trial and appellate courts at both the federal and state level. In general, a lawsuit will be filed with a trial court which will decide the case on its merits. An aggrieved party then has the right to appeal the decision to an appellate court.
At the state level, the system varies, but generally you will have minor courts such as a municipal court (handling issues like traffic violations), major trial courts such as a county superior court (handling significant crimes like felonies), and at the top you will have a state supreme court.
At the federal level, trials are handled in federal district courts, of which there are 94 in the United States. At the intermediate appellate level there are 12 appellate courts, called Courts of Appeals, each representing one of the 12 regional circuits. Finally, the court of last resort is the U.S. Supreme Court.
Whether a case is heard at the state level or federal level depends on rules of jurisdiction, and in some cases could be heard in either arena. Indeed, the U.S. Supreme Court will hear cases that originated in both federal and state courts. The Supreme Court agrees to take only a tiny fraction of cases presented to it, and once it renders judgment there is no further place to appeal.