I'm presuming you know what murder means, so I'll just tell you about the other two.
Assassination is the public killing of a prominent person, usually a politician. The purpose of assassination is to make a political or moral statement.
Homicide is a technical/legal term. It has a broader meaning than 'murder' as it can, for example, include the lesser charge of 'manslaughter', and the non-criminal act of killing someone in self-defence.