State and federal courts are organized under different laws and constitutional provisions. Federalism is one of the founding precepts of the United States and it provides for a separation of state and federal courts. State and federal courts have different organization and structure as well as different kinds of cases that they hear. While both criminal and civil cases can be heard under either a state or federal court, or even both, the jurisdiction to hear cases begins back to the United States Constitution and even before, in the colonies.
Federal courts are organized under specific provisions of the United States Constitution. In fact only the United States Supreme Court is mentioned in the US Constitution. Article III set forth the provision for Congress to establish and ordain a Supreme Court as well as inferior courts as Congress might require from time to time. So besides the Supreme Court Congress could create and, by extension of the "time to time" and "as required" language, eliminate. There is no requirement that Congress create any other federal courts, but the size and scope of the United States legal system has led to creation of federal district courts and circuit courts of appeal to handle a variety of legal issues. Certain special courts are empaneled to hear particular cases like claims against the Unites States government or bankruptcy. Bankruptcy only exists in the federal court system.
Federal courts have jurisdiction over a few different types of cases. The federal courts get jurisdiction from Congress through the US Constitution. The first way federal courts get to hear a case is through what is called "federal question" jurisdiction. In federal question jurisdiction the federal courts have subject matter jurisdiction to hear cases involving some violation of the laws, treaties and constitution of the United States. The second is called "diversity jurisdiction" which arises out of legal conflicts between parties of different states or foreign citizens.
The federal courts do not only handle civil cases, but also criminal cases. While originally the US Constitution only authorized authority in three areas, counterfeiting, piracy and treason, the "necessary and proper" clause made a hole so big that it led to the current swath of crimes enumerated in the United States Code. Similarly, the federal courts hear bankruptcy cases.
Under the federal system, judges are confirmed by the United States Senate after being nominated by the President. Federal judges usually serve a life term in office, but may be removed through an impeachment process.
State courts operate fairly similarly to federal courts. Each state constitution establishes the types of courts. Most states will have a three tier system that roughly mirrors the federal court system. Typically there is a trial level court, an intermediate appellate court and a supreme court. Each state will name these courts different things. For example, the highest court in the State of Florida is called the Supreme Court while the trial level court in the State of New York is called the Supreme Court.
State courts hear criminal cases and administrative cases as well as family law, probate law (wills and estates), tort cases, contract disputes and more. Any dispute involving the laws and constitutions of a particular state are decided by state courts.
State judges are selected very differently from federal judges. State court judges can be elected, appointed, either for a number of years or even life, or a combination of both election and appointment. The number of judges that hear a particular case varies from state to state, but most often a trial court
has a single judge. An appellate court has more than one judge, sometimes 3 or even 5. State supreme courts usually mirror the federal system and have 9 justices, some or all of which may resolve a legal controversy. State courts handle the vast majority of criminal and civil cases in the United States.
There are a few intersections of federal and state courts. If a state court tries to interpret some provision of federal law this may be something that ends up ultimately being decided at the federal appellate level. Also, when a federal court hears a diversity jurisdiction case, although the federal court sits in judgment, it must apply the laws of one or both of the states to the legal controversy as required. Federal courts can also hear appeals and post-conviction relief cases that arise out of state law if some form of US constitutional protection is implicated in the case. For example, even though states often have similar protections to those in the United States Constitution, they may not provide a high enough level of protection.
States are free to provide more constitutional protections than afforded by the US Constitution, but not less. State and federal courts can also end up prosecuting a defendant for the same acts which are criminal under both state and federal law. This is called separate sovereigns and allows for two prosecutions without invoking double jeopardy. Finally, many federal constitutional protections are incorporated to the states through the 14th Amendment to the United States Constitution. It becomes the job of state courts to ensure that those provisions are properly enforced.