There are several power struggles in The Crucible.
Most power struggles find their root in one of a couple of character groups: Abigail Williams or the Putnam family. To understand these struggles, the narrative text that interrupts dialogue explains background information that audience members might not readily know.
Abigail Williams is engaged in a power struggle with John Proctor. This is perhaps the strongest move for power in the whole play. Abigail uses her previous adulterous relationship with John to try to manipulate her way back into his arms. We see this in the first act as Abby tells him how he "sweated like a stallion" for her and how she feels his "heat" below her window at night. Although John confesses to this, he pushes back by telling her he doesn't want that type of relationship anymore. As the play continues Abby makes threats to claim John's wife has seen the Devil. She eventually does, and this power play ends up bringing John and his current servant Mary Warren down to the court. At this point, Mary is on John's side but Abby manipulates her through peer pressure to join with the girls again.
The Putnam family is in a power struggle with the Nurse and Proctor families. This happens because Mr. Putnam believes his father left property to him, but the Nurse family claims the property was actually theirs, not something that could be willed to Mr. Putnam. The Nurse family sold the property to the Proctors. Thus, there is a power struggle between the Putnams and these other families. Many believe that the Putnam's daughter was one of the children who started calling out names of people who had supposedly seen the Devil on behalf of her parents. She was calling out people with whom her parents had much conflict.