Arthur Miller himself does a good job of giving you the background information on the relationship between these two prominent families. If you look in Act One, Miller steps away from the action of the play for a moment, to fill you in on what has been going on in Salem, in regards to the Putnams and the Nurses, and why they have reason to dislike one another.
Francis Nurse, an old, wise man, was often
"called upon to arbitrate disputes as though he were an unofficial judge."
So, he stepped in and voiced his opinion on matters of the town, and people took him very seriously. His weight and pull amongst the townspeople came into play as he was part of the campaign against the guy that Thomas Putnam wanted to be the new minister in Salem. Millers states specifically that
"the Nurse clan had been in the faction that prevented Bayley's taking office."
So, they pretty much prevented the guy that Thomas Putnam wanted to be minister from being elected.
Secondly, the Putnams and the Nurses had fought over land boundaries for years and years. They had neighboring lands, and the fight got so bad that at one point there was "a battle in the woods" that lasted a couple days. Then the Nurse family joined with several others in breaking away from Salem and starting their own community, Topsfield. This was just one further insult to the Putnams, who were so highly regarded, and so much in power, in Salem.
You can find all of this information in the middle of Act One, where Miller breaks away from the dialogue for a while. I hope that helped; good luck!