Framing was the act of making it appear as though somebody other than the culprit had committed a crime, often murder. Motives for framing someone included revenge and deflection of guilt from the actual party.
One could also be framed for a crime which had not in fact occurred. In 2267, Benjamin Finney framed James T. Kirk for his apparent "death" in order to get revenge on Kirk, whom Finney believed to be responsible for his not being promoted. (TOS: "Court Martial")
In 2368, Jev attempted to frame his father, Tarmin, for his telepathic memory invasions of members of the crew of the USS Enterprise-D. Though his attempt was initially successful, it was ultimately determined that Jev was responsible. (TNG: "Violations")
In 2374, Arturis, a member of Species 116, tempted the crew of Voyager with an advanced starship known as the USS Dauntless featuring quantum slipstream drive technology. The ship was supposedly sent by Starfleet, with the details in a transmission sent from the Hirogen communications network. When Seven of Nine and Captain Kathryn Janeway discovered evidence that the ship was not what it seemed, they realized the deception by Arturis, who had supposedly decoded the transmission. When confronted, Arturis tried to frame Seven of Nine stating that she was trying to sabotage their every effort to reach Earth and that Janeway could find the evidence in Seven's personal database. Janeway, however, did not believe the deception, stating that the evidence was undoubtedly placed there by him. She ordered him placed in Voyager's brig, at which point he dropped his deception. (VOY: "Hope and Fear")