It really depends on the state, since criminal law is mostly statutory.
But, here are some generalizations. (I did not google).
Murder, like all crimes, is composed of an actus reus (the act), and a mens rea (the thought).
In common law, murder is “An unlawful killing, with malice aforethought”.
This translates into:
Actus reus- killing which is caused by your actions or inactions which you are responsible for.
Mens rea- one of the following:
Knowing you are killing, or
Knowing you are grievously injuring, or
Acting with extreme recklessness towards someone dying, or
Knowing you are killing or grievously injuring someone else, or
Commiting a felony (only a dangerous one)
Manslaughter, is the same act, but has a lower mens rea. Manslaughter in common law requires only recklessness, or some call it criminal negligence. Or, the commission of a misdemeanor.
Homicide is any killing which is illegal. Criminal homicide sounds like a statutory term, which is used to differntiate among homicides.
First degree and second degree are statutory inventions, and usually depend on factors like, premeditation, the weapon used, what other crimes were being committed, and things like that.