One of the biggest sources of confusion surrounding double jeopardy was the fact that the amendment once applied to federal law. But today, each state uses the Constitution to establish precedent. No standardized version of double jeopardy exists for all the states, so each state can interpret the law differently.
Now, the major confusion comes from determining whether a person is being charged with the same offense. One case, Blockburger V. United States, determined that an offense qualifies for double jeopardy if it does not require additional evidence. One charge must be differentiated from the other.
Additionally, double jeopardy only refers to criminal court cases rather than civil cases. Administrative issues may also come with additional consequences. For instance, the DMV can issue administrative penalties in addition to penalties levied by a criminal court.
You can look at the O.J. Simpson case for a better understanding of how double jeopardy works in the United States. While Simpson was declared not guilty by the criminal courts, civil courts had the opportunity to try the case separately, finding him liable for two wrongful deaths.
Additionally, two separate sovereign states can try an individual for the same crime. Simply because an individual was acquitted in state court does not mean they cannot be tried in a federal court as well.