Hi!
First degree murder is when the murder is premeditated. Essentially, the prosecutor has to prove that you thought of committing the crime before it took place, through planning etc.
Capital murder: More or less what you call "First degree murder" in states where the death penalty is legal.
Felony murder is what you would charge a suspect with if he/she killed someone while committing another crime such as a robbery. If the intent was not to kill during a mugging, for instance, you'll probably get charged with Felony murder, instead of 1st degree murder. However, it still counts as murder in the first degree.
Second degree murder:
A killing done impulsively without premeditation, but with "malice aforethought"
A killing that results from an act intended to cause serious bodily harm
A killing that results from an act that demonstrates the perpetrators "depraved indifference" to human life (Source)
Here's an example: Say you come home to find your wife sleeping with another man and then you let him walk away. The next day, however, you see him driving past you on the street. You take out your gun and kill him, spontaneously.
As we can see in this situation, there is no premeditation/planning involved, the crime was not committed in a "heat of passion", but it was intentional - making it a murder. You also recklessly fired your gun from the sidewalk at the car in the road, showing no regard for potential people behind the car, which proves your "depraved indifference to human life". Killing someone behind the target you intended to kill in this situation would also count as murder in the second degree.
(There is also third degree murder, but it's only used in a few states so I won't explain it)
This brings us to another common criminal charge which is Manslaughter.
Voluntary manslaughter occurs when someone with no prior intent to kill, kills someone in a fit of rage or maybe because of a fight.
Involuntary manslaughter is what you might get charged with if you kill someone by accident through reckless behavior.
Moving on...
An assault does not cause physical harm to the victim while battery does. If someone raises their fist towards you but misses and doesn't harm you, that person has committed an assault, but not battery. Battery requires an "offensive" or "harmful" physical touch. There's also simple assault vs. aggravated assault.
An aggravated assault is an assault that someone commits with the intent of committing a more serious crime, such as rape.
I believe Assault with a Deadly Weapon can count as both assault and battery, regardless if a victim was harmed or not. It makes no difference if someone threatens you with a golf club or if someone hits you with it - same criminal charge. A prosecutor just has to prove that the weapon was capable of causing death.
DISCLAIMER: I'm not a lawyer, don't use my definitions for your real-life criminal defense.