Quite simply, Jim Crow laws are the laws of racial segregation in the United states that existed from just after the Civil War up until the mid-sixties. They were supposed to insure "separate but equal" public facilities, but basically assured that whites got the best of everything. (I'll never forget the double water fountain at my high school, one with cold water and one with warm water: an ancient Jim Crow antique.) Most everything was affected by Jim Crow laws: all public places (transportation, schools, bathrooms, drinking fountains, ... even the military!).
The more interesting part of your question is "how do they relate to To Kill a Mockingbird," of course. Well, considering the dates of the Jim Crow laws, you will notice that they exist avidly in the context of the setting of this fine novel. Although many of Maycomb's citizens are quite happy with the Jim Crow laws, Atticus is not.
Atticus protests these laws quietly, ... and not so quietly, ... in his defense of Tom Robinson. He certainly doesn't allow Scout or Jem to use racial slurs. Another character that is imperative to this discussion would be Calpurnia. Atticus regards Calpurnia as an equal. She doles out wisdom and she mothers both Scout and Jem. Atticus isn't afraid to stand up to people who disagree, either, ... such as Aunt Alexandra. In this way, Calpurnia can be regarded as a very prominent member of the Finch family! In fact, it is from this high perch that the children observe the very trial where Atticus tries desperately to symbolically defeat Jim Crow.