In "The Wife of Bath's Tale," what agreement does the knight make with the old woman?
As the other excellent answer to this question suggests, the knight agrees to grant the old woman the first thing she desires if she gives him the answer to the riddle regarding what women want. Again, what the old woman desires is the knight's promise to marry her. This turn of events is significant in many ways, especially since the woman's answer to the question of what women want is "sovereignty," and, as it turns out, this sovereignty is especially important in the context of marriage. Thus, the knight's agreement becomes an illustration of men submitting to women's desires and allowing them sovereignty, especially in relationships. Indeed, the knight not only gives the old woman what she wants (marriage), but he's also providing her with the thing that women, according to the Wife of Bath, universally want: power over men. It's a radically feminist notion for a text as early as The Canterbury Tales, but this radicalism is tempered somewhat by the old woman's decision to please the knight by becoming young and beautiful. This decision is especially troublesome once we remember that, at the beginning of the story, the knight raped a woman. With this idea in mind, does he really deserve to be rewarded?
Let us just briefly recap: the knight, driven into a desperate state by his unsuccessful search to the answer to the question posed to him by the queen, comes across the old woman the night before he has to give his answer or else pay the forfeit. He throws himself upon the mercy of the old woman, who agrees to help him, but only if he agrees to do something for her in return. Consider what she says to him:
"Put your hand in mine and pledge your word," said she,
"That you will do the first thing I require
Of you, so be that it lies in your power,
And I shall tell it to you before night."
Thus the knight agrees, desperate to find the answer and save his life. It is only later that the old woman reveals what the thing she requires of him will be: his marriage to her. This of course allows the Wife of Bath the opportunity to display how the knight is forced not just to intellectually know what it is that women desire, but to live it out in his marriage as well.