The Fifth Amendment of the U.S. Constitution prohibits the same government from trying a defendant twice for the same offense: "... nor shall any person be subject for the same offence to be twice put in jeopardy of life or limb ...."
The origins of this prohibition are in English common law. Sir William Blackstone declared in his Commentaries on the Laws of England that it was a "universal maxim of the common law of England, that no man is to be brought into jeopardy more than once of the same offense." The principle also existed in Greek, Roman and canon law.
Beware, however, is that there are many surprising exceptions in American law that sometimes do apparently permit trying someone twice for the same offense. For example, a defendant can be acquitted in a state prosecution yet retried and convicted in a federal prosecution for the same conduct, and vice-versa. The reason for allowing this that federal and state government are different sovereigns. Another exception to the double jeopardy protection is that civil lawsuits can proceed despite a criminal acquittal.
A given criminal conduct can violate different criminal statutes. So, the same person can be prosecuted for more than one offense. For example, a person might kill a bank teller while robbing a bank and then fail to pay taxes on the money taken from the bank, which could be three separate offenses. Also, a criminal prosecution must reach a certain stage before "jeopardy attaches". If the government drops a criminal case early, it can restart a prosecution of the same person for the same offense without "double jeopardy."