A person who has been acquitted of a crime and is tried by the same government for the same crime is being subjected to double jeopardy. In the United States and in many other countries, this is not permitted. In the United States, the Fifth Amendment of the Constitution prohibits this. However, as in most matters legal, this is not quite as straightforward as one might think.
First, this pertains only to criminal matters, not to civil ones. Whether or not one might be sued twice for the same reason is a different matter entirely.
Second, there must have been a complete trial and disposition for double jeopardy prohibition to apply. A hung jury does not protect anyone from being tried again. Nor does some disruption of the judicial proceedings.
Third, in the United States, while most criminal law is state law, there is another layer of law at the federal level and different states are considered different governments in this context. Since these are scenarios involving two different governments, double jeopardy does not necessarily apply.
Fourth, many criminal acts are also what we call torts, civil harms against people or property. One criminal act can result in a criminal proceeding and a civil proceeding. An example of this is the trial of O.J. Simpson, who was found not guilty of murder but was then sued by his wife's estate in a civil proceeding, a proceeding in which he was found liable for her death. This was possible because the standard for guilt is "beyond a reasonable doubt," but the standard for liability in many states is that there is "substantial evidence" against the defendant. That is a much lower standard, so a jury found him liable, the criminal verdict notwithstanding.