What is the marital problem between John and Elizabeth involving their former servant Abigail?
John Proctor had something of an affair with Abigail and Elizabeth found out. This led to Abigail's release from employment in the Proctor household, but Elizabeth and John Proctor tried to keep the story private. Despite their care, suspicion and rumor spread and the town rather accurately surmised the reason that Abigail was fired - blaming Abigail, by and large, for moral failings.
John's dalliance with Abigail is discussed when he is alone with Abigail in the first episode of the play. In this part of the play, Abigail expresses her desires to rekindle things with Proctor. He refuses to entertain her advances.
Proctor: Abby, I may think of you softly from time to time. But I will cut off my hand before I’ll ever reach for you again. Wipe it out of mind. We never touched, Abby.
Proctor's affair with Abigail is discussed again when he returns home to Elizabeth, who he feels is continuing to treat him coldly.
"Although her husband has admitted his lapse into sin and is thereafter faithful to his wife, his relationship with Abigail always stands between them" (eNotes).
The situation of mutual rebuke between John Proctor and Elizabeth (he rebuking her for being unforgiving and she rebuking him for being unfaithful) generates one of the conflicts of the play. There is a tension and a hope for conciliation between the two that is complicated further when Elizabeth is accused of witchcraft and arrested.
Abigail's motives for accusing Elizabeth and conducting a thought-out ruse to have Elizabeth arrested stem from her feelings for John Proctor and her jealousy of Elizabeth.