Double jeopardy is a situation in which a person is tried for a crime, is acquitted, and is then tried again for the same crime. It can also refer to a situation in which a person is punished more than once for the same crime. Double jeopardy is banned by the Constitution of the United States.
The ban on double jeopardy is an important protector of our rights. Without it, the government could persecute us repeatedly for no reason. For example, imagine that the government suspects you of being a terrorist. It tries you for a crime and you are acquitted. If there were no protection against double jeopardy, the government could simply keep trying you over and over for the same crime. Your life would be completely ruined even though you had, presumably, not done anything wrong.
We should note that double jeopardy does not apply to suing someone who has already been acquitted in a criminal trial. That is why, for example, OJ Simpson could be sued for wrongfully killing someone even after he had been acquitted of their murder in a criminal trial.