First degree murder is usually premeditated, while second degree murder is either murder that was not premeditated, or murder that was the result of obvious lack of concern for human life. There is also voluntary manslaughter which is murder that happens in the "heat of the moment"
Example 1: Tom keeps spreading rumors about Bill. Bill gets pissed off and start making plans to kill him. Bill convinced Tom to meet him at the lake for some late night fishing. Tom is found washed up on shore a week later. Bill is eventually caught, tried, and sentenced to life in prison for premeditated murder (1st degree).
Example 2: Tom and Bill get in a very heated argument. The next day, Bill and Tom accidentally meet up again, but this time Bill kills Tom. Since the murder was not planned, Bill is serving a 20 to life sentence for 2nd degree murder.
Example 3: Tom and Bill are in an extremely heated argument. In a moment of pure anger, Bill hits Tom over the head with a lamp. Tom dies. Bill ends up serving 10 years for voluntary manslaughter since the crime happened in the "heat of the moment".
EDIT: Oop, c3 beat me to it. I guess I took too long.