The climax is the most emotional moment of the plot and also the moment when the resolution comes into view. As the title suggests, since the plot is about two parallel and intertwining stories of two sisters, one who governs her actions with sense and one who governs her actions with her sensibilities, there are actually two climactic moments in the story.
The first climax takes place when Marianne receives her heartbreaking letter from Willoughby. It is at this moment that the reader knows for sure that there is absolutely no future relationship for Marianne and Willoughby. In fact Willoughby's letter, which we learn much later was not actually written by him, so cruelly apologizes if he had made her believe he felt more than "esteem for [her] whole family" (Ch. 29). He further states that he has long been engaged and returns both her letters and lock of hair, which for the culture of the time, both happen to be tokens of an engagement. Men and women did not exchange locks of hair or letters at that time unless they were engaged. Hence, we see it is at this point that we learn the true nature of Wickham's character as well as the true nature of his relationship with Marianne. At this point, the plot begins winding down. Though Marianne's heartache nearly leads to her death, her near death experience resolves her beliefs that emotions should always be guarded and leads instead to a belief that agrees with her sisters, that there is a need to use sense in all things, including governing one's emotions. It is also Marianne's heartache and near death experience that lead to her relationship with Colonel Brandon, tying up that lose end as well. Therefore, the climax involving Marianne definitely occurs when she reads Willoughby's letter.
The climax in the plot involving Elinor happens a bit later. At first we have a false climax when Fanny Dashwood learns about Edward's secret engagement and it begins to look like his marriage to Lucy is inevitable. However, no resolution is really established at that point. Therefore, the real point of climax happens when, just after the Dashwood servant, Thomas, announces he has seen Miss Lucy Steele with her new husband, Edward appears at Barton Cottage and announces that Lucy has actually married his brother. It is at this moment we know that a marriage between Edward and Elinor is inevitable, creating our second resolution, as Austen explains, Edward's "errand at Barton, in fact, was a simple one. It was only to ask Elinor to marry him" (Ch. 49). In addition, this is an extremely emotional moment for Elinor, but for Edward as well. When Elinor learns the news, she is so overcome with tears of joy that she has to leave the room. Edward as well, when he sees Elinor's emotion, must stroll outside in order to regain his composure. Hence, this moment in the plot is a tensely emotional moment, leading to the resolution, making it a climatic moment.