The Tenth Amendment of the U.S. Constitution mandates a division of powers between the federal government and the state governments. This means that there are separate judicial systems that deal with different types of cases (or jurisdictions). Generally, the federal court system only has jurisdiction over cases that arise from federal laws, or that affect the entire country or federal government, or certain cases that involve more than one state. The remainder (including most criminal cases) falls under the jurisdiction of state courts.
There are 94 district courts across the country. Some states have only one district (e.g. Massachusetts), while others have more than one (e.g. California has four: Southern, Northern, Eastern, and Central).
Appeals courts, which hear appeals from the district courts, are arranged into regional circuits, each of which has the jurisdiction over a number of district courts. For example, the First Circuit hears appeals of cases from the District of Massachusetts (along with Maine, New Hampshire, Rhode Island, and Puerto Rico), and the Ninth Circuit hears appeals from the four districts of California (plus a number of other states of the west coast).
The Supreme Court is the highest court in the U.S., and hears appeals from both federal and state appeals courts.
State courts have at least two levels of court: the trial court(s), and the appeals court(s). Depending on the state, there may be other types of court. The Massachusetts court system consists of the Trial Courts (which is further divided depending on the type of case), the Appeals Court, and the Supreme Judicial Court.
The Wikipedia article List of courts in the United States outlines the structure of each state's court.
In addition to general courts, there are a number of specialty courts (or tribunals) that only hear cases in a certain subject matters, for example, tax, bankruptcy, family law, patents, military law, etc.
Trial courts are the first court that will hear the case. These cases are heard on both the facts and the law. Sometimes a jury of peers will be the "triers of fact", who determine whether things happened the way it is claimed and whether someone is guilty (in a criminal trial) or liable (in a civil trial). They may also determine the sentence or damages. The judge is the "trier of law", who determines what laws apply to the situation and how they must be interpreted.
If a decision is appealed it goes to an appellate court. Appellate courts only hear arguments on issues of law; they do not rehear the facts of the case and they do not include new facts. There may be more than one judge hearing an appellate case. A decision of the appellate court could be to affirm the lower court's decision, overturn it (or part of it), or remand (send it back to the lower court for another hearing).