What is the difference between first-degree murder second-degree and third-degree murder?

Each jurisdiction (in the US, that basically means "state") sets its own definitions. The general concept is that lower "degrees" are more "serious" and punished more harshly.

A very common differentiation is the first-degree murder ("murder in the first degree", "murder one") usually is defined such that premeditation is required. In order to commit murder one, the killer must have planned it in advance. A murder committed "in the heat of the moment" using the first implement that came to hand would usually not be murder in the first.

States may also use different terms like "capital murder" (a murder which meets the criteria for a possible death penalty sentence" vs. simply "murder" vs. "manslaughter" (killing a person under circumstances that, for some reason or other, do not warrant the term "murder").

You can consult your state's criminal code if you'd like to know exactly how they're defined where you live.