In The Crucible, by whom is the following said, "The promise that a stallion gives a mare I gave that girl."
That quote is spoken by John Proctor. He is talking to his wife about what he did or did not promise to Abigail Williams.
What he means by this quote is that all he was doing was having sex with Abigail -- that there was no emotional commitment. He did not promise her love or marriage anymore than a stallion promises a mare love or marriage when they mate.
This argument happens in Act II, Scene 1. Elizabeth Proctor is telling her husband that he must break his promise to Abigail. She says Abigail clearly believes that John loves her and has promised to love and/or marry her. It is at this point that John speaks the words you cite.
John Proctor makes this remark to his wife Elizabeth some time after their maidservant, Mary Warren, has returned from the village where she has become an official of the court during the witch trials. Mary has informed them that she had just saved Elizabeth's life. She states that Elizabeth had been accused and that she defended her. She refuses to divulge the name of Elizabeth's accuser.
Elizabeth suspects that it is Abigail Williams, their erstwhile maidservant, who accused her. Abigail had been ignominiously dismissed by Elizabeth when she discovered that she and her husband were involved in an adulterous affair. Elizabeth fears that Abigail has accused her out of vengeance and wants her dead, saying: 'The noose is up!' She fears that Abigail wants her dead so that she may reunite with John.
It is important to note that Elizabeth and John had been having a verbal altercation just before Mary's arrival. They were arguing about Elizabeth's suspicion that John had been lying to her and that he still had a soft spot for Abigail. John was quite angry about Elizabeth's suspicions and wanted her to not only forgive him, but believe that he had completely dismissed Abigail from his heart and his mind. it is obviously difficult for Elizabeth to do so because he had admitted that he had spoken to Abigail alone whilst he had previously stated that they had spoken when in company. John bitterly resents Elizabeth's judgment of him.
When Mary Warren has gone to bed, Elizabeth beseeches John to go to court and testify that Abigail is a liar and a whore. When John expresses doubt that he would be believed, Elizabeth insists, saying that if it had not been Abigail specifically against whom he had to testify, his attitude would have been different. She furthermore requests that John go directly to Abigail and speak to her. John is recalcitrant and Elizabeth then says:
You have a faulty understanding of young girls. There is a promise made in any bed -
John reacts angrily, demanding to know what promise.
Spoke or silent, a promise is surely made. And she may dote on it now - I am sure she does - and thinks to kill me, then to take my place.
Elizabeth tells John that she believes Abigail wants to profit from her death by replacing her. She picks on John telling him about how he blushes when they see Abigail in church. John becomes angrier by the minute. Elizabeth wants him to identify Abigail as a whore, out for vengeance. Elizabeth believes that John is ashamed and that is why he blushes, but that Abigail may interpret it differently.
Elizabeth tells John:
Then go and tell her she’s a whore. Whatever promise she may sense - break it, John, break it.
John says angrily that he will do as she asks but when she says he seems unwilling, he rants that Elizabeth should not deny him his anger and that she thinks him base. He demands to know how Elizabeth can accuse him of making a promise to be with (marry) Abigail. It is at this point that he shouts out the words mentioned in the quote:
Then how do you charge me with such a promise? The promise that a stallion gives a mare I gave that girl!
What he means is that there had been no promise, his affair with Abigail was born out of lust and nothing else. He uses the animal imagery to indicate the senselessness of Elizabeth's accusation. Elizabeth, however, does not back down, asking John why he becomes so furious when she asks him to break his 'promise'.
John reacts by saying that Elizabeth's request speaks of deceit and that he is honest. He asserts that he now sees that Elizabeth will never forgive him but that he is done pleading for her forgiveness. Elizabeth cries out:
You’ll tear it free - when you come to know that I will be your only wife, or no wife at all! She has an arrow in you yet, John Proctor, and you know it well!
She states a clear threat and concludes that her husband knows full well that Abigail still has some influence over him. It is at this point, as if by divine intervention, that Reverend Hale arrives, putting an end to their devastating quarrel.