The much-publicized Oklahoma City bombing cases provide relevant examples. The two defendants in that crime were convicted in federal court. Timothy McVeigh was given the death penalty and subsequently executed. The other defendant, Terry Nichols, was also convicted in federal court and given life imprisonment with no hope of parole. He was later tried in an Oklahoma state court, convicted of 161 state murder charges, and sentenced to life times 161.
This did not constitute double jeopardy because of dual sovereignty. Defendants can also be tried in two different states for essentially the same crime, if the crime or an element thereof was committed in those states. The cases of John Allen Muhammad and John Lee Malvo, the two snipers who terrorized the Maryland–Washington, D.C.–Virginia areas in October 2002, provide another example. Accused of shooting 19 people and killing 13, they were tried and punished in federal court as well as in state courts in places where the shootings and other crimes took place.
Whether the state can and will try a defendant again depends on state law and the discretion of the prosecutor. The government that first obtains custody of the suspect is usually allowed to try him or her first. In most cases, this is the state. Although the federal government can try the defendant for the same offense, it usually refrains from doing so if the defendant has been convicted and sufficiently punished under state law.