How does the federal court system relate to the state court systems?
Generally speaking, the federal court system and the state court systems run along separate tracks, with very different jurisdictions, but there are occasions upon which federal courts hear state claims and upon which a state may hear a federal claim. The jurisdiction of the federal courts includes cases involving the United States Constitution, all federal statutes and regulations, and United States treaties. Additionally, cases involving multiple states, for example, one state suing another, are under federal court jurisdiction. The jurisdiction of a state court is over all cases arising under state law, regulation, or state constitution. Sometimes federal courts will hear a state claim when there is a federal claim, too, and the cause of action arises from the same occurrence. For example, a person might allege unlawful discrimination under the Civil Rights Act, which is a federal statute, and also attach a claim under a state anti-discrimination statute. The underlying occurrence is the same, and the federal court has jurisdiction to rule under both statutes. Similarly, a state court may hear a federal claim, if the Constitution has not provided that the federal courts have exclusive jurisdiction.
As a general matter, most attorneys are admitted to practice before their state courts and also gain admission to practice in federal court, in the circuit in which their particular state is, at the very least. In order to be a good "full service" attorney, it is best to be able to practice in both systems.