Jurisdiction is the power of a court to hear and decide cases. In criminal law, it includes the power to impose punishment. Absent jurisdiction, convictions and court-ordered sentences are void.
(For an overview of some basic jurisdiction terminology, see What are the kinds of criminal jurisdiction? And, to learn about the relationship between jurisdiction and venue, see Venue.)
In general, federal courts have exclusive jurisdiction over federal offenses, and state courts have exclusive jurisdiction over state offenses. (But some kinds of conduct qualify as both state and federal offenses—see State vs. Federal Prosecution.) Federal criminal jurisdiction commonly arises where:
Federal courts may also have jurisdiction over crimes occurring in foreign countries if the defendant intended an effect to occur inside the U.S.; examples are cybercrimes and terrorist attacks. But, depending on the foreign country, the federal government may have difficulty hailing the offender to court.
Federal jurisdiction may also extend to offenses occurring on Native American reservations. Whether there's federal jurisdiction may depend on the seriousness of the crime and agreements between the relevant tribe and other governments.
Most crimes that occur within a state's borders, or within three miles of its coastline, are prosecuted in state court. But some federally owned land rests entirely within one state's borders—if a crime occurs on that land, then it's typically a federal offense.
State law usually specifies which courts have jurisdiction over which types of cases. In many states, adult felonies and misdemeanors are divided between different courts. Most states also have juvenile courts, which have exclusive jurisdiction over crimes committed by minors. (Juveniles can, however, be tried as adults in some circumstances.)
The names for the kinds of state courts vary from state to state. Courts of general jurisdiction are usually named "superior" or "circuit" courts. "Municipal court" is a common kind of limited-jurisdiction court.
Concurrent jurisdiction exists where more than one court can claim power to decide a case. For example, more than one state might have jurisdiction where the crime begins in one state and continues into another. Or a crime might occur partly on federal land and partly on non-federal land, giving the state and the federal government jurisdiction. (See If a crime occurs in two or more states, can each prosecute it? Also see State vs. Federal Prosecution.)
Often, the first court to exercise jurisdiction over the defendant keeps the case to itself—at least for a while. That state's jurisdiction will be exclusive; any other courts claiming jurisdiction must wait until the case is over or the first court releases its jurisdiction. Merely filing a charge isn't enough to obtain jurisdiction—the government must get the defendant to court. (It usually does so by arrest and arraignment).
Some state laws and constitutions address concurrent jurisdiction, but others don't. In most cases, the first-in-time rule applies. When there are competing claims to jurisdiction, the prosecuting attorneys usually get together and try to reach an agreement about who will "go" first.