To assassinate someone means to target and deliberately kill that person. The word assassinate is almost always used to describe the killing of a famous person.
When a political leader or very well-known person is murdered, the killer is said to assassinate the victim. A killer might assassinate someone for political or ideological reasons, to gain fame for the act, or possibly for payment by someone else. You can assassinate figuratively, as well, by damaging a person's good name: "If you print that story, it will assassinate my reputation." The Arabic root of assassin is hashishiyyin, "hashish users."