According to the Necessary and Proper Clause of the U.S. Constitution, Congress possesses the authority to create laws that categorize certain acts as criminal and to define the punishment for these crimes. State legislatures also have a role in creating state criminal law. However, the criminal laws that state legislatures pass cannot contradict the national or state constitution. The laws that states pass must also protect the welfare of the larger society.
Some crimes are considered state crimes while others are federal crimes. Federal crimes contravene federal law; examples include fraud, counterfeiting, immigration law offenses, postal offenses and possession of illegal substances. State crimes include traffic offenses, illegal possession of firearms and murder. But interstate crimes may also fall under the jurisdiction of federal criminal law. Additionally, if a crime is both a criminal and federal crime, such as the illegal possession of firearms, then federal criminal laws are applicable. The Supremacy Clause of the Constitution enables federal criminal laws to overrule state criminal laws in such instances.
Dual sovereignty enables the federal and state governments to apply both state and federal criminal laws to to a similar criminal act. State and federal criminal laws are sovereign is cases where the state and federal governments have an interest in prosecuting the same criminal act. The same crime may be prosecuted under state and federal laws without inflicting double jeopardy on the convicted person. Such crimes that invoke dual sovereignty in criminal law include heinous acts such as the murder of a congressperson.
The state government is usually responsible for enforcing criminal laws -- even when an individual has committed a federal crime. The reason is that residents of a state are considered under the jurisdiction of that state. On the other hand, the federal government's role has been to police and regulate federal and interstate crime. The Eighth Amendment prescribes the enforcement of both state and criminal offenses.