It differs from place to place, but the premise is the same. The main determiner in degrees is how much malicious intent motivated the action.
First degree is the most serious, because it indicates that a person calmly decided to kill another person, decided how to do it, and carried out their murderous plan. These are your intentional murders, especially where the killer lies in wait or plots to kill. These are typically the capital form of homicide (viz., they can result in the death penalty).
Second degree homicide is less intentional, but it is intentional nonetheless. These can include unforeseen brawls where someone pulls a weapon, where a spouse comes home to find their partner being unfaithful, or where an ordinarily peaceful person is provoked to suddenly lash out.
Third degree homicide usually includes killing that is basically unintentional but that is so negligent that society punishes it. This could include something like a driver who speeds recklessly, loses control, and kills someone else on the roadway; the driver did not intend to kill anyone and probably did not even know the victim, but he was intentionally neglecting laws that are meant to keep people safe on the roadways and caused a death through that law-breaking negligence.
Some states break down the degrees a little further. Most states also make a distinction between homicide, which is more intentional, and manslaughter, in which there is less proof of malicious intention.
Manslaughter - you accidently killed somebody
1st Degree - Premeditation -- you planned to kill somebody
2nd Degree - You killed somebody willfully but didn't plan to do it (e.g. crime of passion)
Manslaughter - you accidently killed somebody
Negligent Homicide - you killed somebody but becuase you were an idiot. By action or inaction you allowed somebody else to get killed.
I forgot one-- Justified Homocide. -- you killed somebody in self defense.
1st Degree - Premeditation -- you planned to kill somebody
2nd Degree - You killed somebody willfully but didn't plan to do it (e.g. crime of passion)
Manslaughter - you accidently killed somebody
Negligent Homicide - you killed somebody but becuase you were an idiot. By action or inaction you allowed somebody else to get killed.
I forgot one-- Justified Homocide. -- you killed somebody in self defense.
