The motivation for the most Abigail's actions were based on her adulterous affair with John Proctor. According to the play, she wanted to get rid of Elizabeth Proctor and then marry John. Elizabeth also implies some sexual repression when she mentions to John that she kept a "cold house" and that might have been partially responsible for John's actions. During the play, John rejects Abigail's affections and is obviously sorry for the affair. Abigail, obviously hurt by John's rejection, wants revenge but also something to help raise her self-esteem after being cast out of the Proctor household. She is able to become the focus of the town during the first part of the trials.
Finally, Elizabeth refuses to acknowledge John's affair with Abigail in front of the court, thinking she is sparing his reputation, Ironically, John has already admitted to the affair so her refusal to go public about it does nothing to help the cause of either her husband or the other accused of witchcraft.