To accuse someone of doing something unlawful in a court of law or to try to prove in a court of law that a person is guilty of having committed a crime. Both the person that accuses a person of a crime and the lawyer working for the plaintiff/ district attorney are trying to prosecute the defendant. This term is normally used in criminal law cases as there is a link to crime and prosecution but a personal can be prosecuted in civil court as well.
A criminal defendant is entitled to due process which means the benefit of criminal proceedings carried out in accordance with the governing law of the land. Within the criminal proceeding it is mandatory that the point being made is proven beyond all reasonable doubt and the act of doing so is the prosecution, the act of doing so is to prosecute.