What is the relationship between Abigail and John Proctor in The Crucible by Arthur Miller?
The short answer to your question is that Abigail Williams and John Proctor are ex-lovers. We discover in the first act of The Crucible by Arthur Miller that the two of them had an affair. Abigail was the Proctor's servant girl, and it was, at least according to her, a very intense affair.
We learn that the affair ended seven months ago; however, Abigail makes it clear that she wants Proctor back. Proctor, on the other hand, is determined not to resume their relationship (which is just a bit different than saying he does not still want to be with her). Proctor says it this way:
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.
John Proctor wins this battle of wills and he does not resume his relationship with Abigail Williams. Nevertheless, this relationship is the crux of all that goes wrong in Salem during this witchcraft frenzy. It seems to me that you have not read very far in the play yet, so I won't spoil anything for you; however, I would encourage you to keep this relationship in mind as you read. Abigail is a liar (Miller says she has "and endless capacity for dissembling") and she is desperate to get her lover back. It is a deadly combination.
As the play begins, the audience understands that John and Elizabeth Proctor, a couple in their thirties, had employed Abigail Williams, a seventeen-year-old girl, as household help while Elizabeth recuperated from a difficult birth. Abigail and John have engaged in an affair, and Elizabeth has found out about it and dismissed Abigail, and (according to Abigail) quietly put out the word in Salem that she should not be hired by other families-- without detailing what happened.
Abigail has mistaken John's sexual interest in her for genuine love. She is desperate to win him back; she believes that if Elizabeth were out of the way, she would become John's second wife. This is the reason she was conjuring a spell in the woods with Tituba and girls from the village.
John is committed to restoring his relationship with Elizabeth, and though he is unsettled by Abigail's attraction, vows that it is over between them.