A wealthy American man named Longmore is introduced to his countrywoman Euphemia de Mauves, wife of the Comte Richard de Mauves. Longmore and Madame de Mauves become friends, and he visits her frequently in Paris. Superficially, Madam de Mauves leads a happy life with a wealthy and "irreproachably polite" husband, but Longmore soon becomes convinced that she harbours a deep sadness. It gradually becomes clear that the Comte is an unscrupulous and dissipated man who married his wife for her money alone. As a youth, Madame de Mauves had been naive and idealistic, believing that the Comte de Mauves' title guaranteed a fine character. The Comte, however, proved to have little regard for his wife, and had embarked on a series of extramarital affairs. Even his politeness "was hardly more than a form of luxurious egotism, like his fondness for cambric handkerchiefs.... In after years he was terribly polite to his wife." Madame de Mauves' faith in her ideals is destroyed, but she responds with stoic resignation.
Longmore falls in love with Madame de Mauves, but, understanding that he cannot be her lover, and believing that she desperately needs a friend, he tries to sublimate his love into friendship. This attitude is reinforced by Madame de Mauves, who welcomes his friendship, but is hostile to any sentiment on his part. However both the Comte de Mauves and his sister, the crass widow Madame de Clairin, hint that Longmore should woo Madame de Mauves. The Comte wishes her to take a lover so that he may be free to pursue his own affair.
As tensions mount, the Comte openly breaks with his wife, Madame de Clairin urges Longmore to woo Madame de Mauves, and then she tells Madame de Mauves what she has told Longmore. Longmore agonises over how to proceed; he finds it difficult even to decide to continue his daily visits: "His presence now might be simply a gratuitous cause of suffering; and yet his absence might seem to imply that it was in the power of circumstances to make them ashamed to meet each other's eyes." Eventually he visits Madame de Mauves, who rather cryptically asks him to confirm her very high opinion of him by doing the proper thing: "Don't disappoint me. If you don't understand me now, you will to-morrow, or very soon. When I said just now that I had a very high opinion of you, I meant it very seriously. It was not a vain compliment. I believe that there is no appeal one may make to your generosity which can remain long unanswered. If this were to happen,—if I were to find you selfish where I thought you generous, narrow where I thought you large, ... vulgar where I thought you rare,—I should think worse of human nature. I should suffer,—I should suffer keenly. I should say to myself in the dull days of the future, 'There was one man who might have done so and so; and he, too, failed.' But this shall not be. You have made too good an impression on me not to make the very best. If you wish to please me forever, there's a way."
After much reflection, Longmore concludes that she wishes him to voluntarily break off contact — to do so not because she has dismissed him, not because there has been a 'scene', and not with any promise of meeting again in future, but simply because it is the honourable thing to do. The next day Longmore leaves for America. At his last meeting with the Comte he receives the impression that the Comte may be starting to repent of his behaviour; Longmore feels threatened by this: "he felt that it would be far more tolerable in the future to think of his continued turpitude than of his repentance."
Longmore remains in love with Madame de Mauves, despite having had no contact with her. Two years later, he hears that the Comte has committed suicide. The Comte had indeed repented, and had begged his wife to forgive him, but Madame de Mauves had remained as stoically unforgiving as she had been stoic in her resignation: "[H]e fell madly in love with her now. He was the proudest man in France, but he had begged her on his knees to be readmitted to favor. All in vain! She was stone, she was ice, she was outraged virtue. People noticed a great change in him: he gave up society, ceased to care for anything, looked shockingly. One fine day they learned that he had blown out his brains."
Euphemia is now free, and Longmore's first instinct is to go to her. However he puts off leaving for Europe from day to day for several years, because "The truth is, that in the midst of all the ardent tenderness of his memory of Madame de Mauves, he has become conscious of a singular feeling,—a feeling for which awe would be hardly too strong a name."

Question: What does Comte beg Madam de Mauves for?
Answer: To forgive him.