There are numerous personal grudges in the play, and actually is one of the causes of the witch trials.
The most central person rivalry is between Abigail and Elizabeth Proctor. Abigail had an affair with Elizabeth's husband John. John ended the affair, but Abigail is not willing to take no for an answer. Her jealousy for Elizabeth's role in John's life leads her to accuse Elizabeth of witchcraft.
Other grudges occur between very prominent members of society. Francis and Rebecca Nurse are of a family with lots of children and grandchildren. They own over 300 acres of land, but Thomas Putnam, another prominent member of Salem, disuptes with him over land boundaries. Francis also went against Thomas Putnam's brother in law for the position of minister to Salem.
It is quite a conicidence that Ruth Putnam is an accuser of Rebecca Nurse later in the story. There also seems to be an insinuation that the Putnams are jealous of Rebecca Nurse's numerous children. Mrs. Ann Putnam herself miscarried and lost many of the children she bore.
The Coreys, Giles and Martha, are an older couple. Martha Corey is accused of witchcraft because she sold a pig to a neighbor and the pig died two days later. Since then, the neighbor cannot keep a pig alive on his farm.
There is also a personal rivalry between John Proctor and the Reverend Mr. Parris. Proctor takes issue with Parris's priorities, taking offense at Parris's need to have golden candlesticks on the altar in the meetinghouse when there were already perfectly fine pewter candlesticks made by a member of the congregation. He also says that he doesn't see "the light of God" in the minister and has therefore opted not to have his youngest son baptized: the pair have locked horns over everything from the deed to Mr. Parris's house (which is only ever loaned to the minister of the town for his tenure there) to Mr. Parris's salary and provision for firewood. For his part, Mr. Parris feels that Proctor leads a faction against him in the church, and he insists that Proctor opposes him for no good reason. He is also bitter about his small salary and believes that Proctor is unreasonable for expecting him to spend any of it on firewood when firewood could be given to him in addition to his salary.