Murder is the crime of intentionally killing a person. In legal speak, the element of intention is referred to as ‘malice aforethought’.
“The gunman was charged with murder.”
Manslaughter is the crime of killing a person by a person who e.g. did not intend to do it or cannot be held responsible for his or her actions.
“She was found guilty of manslaughter on the basis of diminished responsibility.”
Homicide is the act of killing someone and does not necessarily denote a crime.
“The carrying out of the death sentence is a homicide.”