Well, because you use the word "scene" in your question, I will assume you are speaking about the film as opposed to the book (not to say the former overshadows the latter, because it does not). There is nothing like the knot in the pit of my stomach at the very end of this movie when Mr. Thackeray (played by the ever-prestigious Sidney Poitier), after being mocked by some future students, exibits determined perseverance, ripping up that job offer and, in so doing, silently vows to stay, continuing to make a difference in students' lives. Those little ripped-up pieces of paper mean more than a lifetime of novels. It's a similar knot (exactly the same, maybe?) as the knot I get when Mr. Keating's students stand on their desks in his honor at the end of Dead Poets Society.
This is clearly a very subjective question with a vast number of possible answers. Obviously you could get a large range of responses. For me, though, when I first read the novel, my favourite scene came in Chapter Eleven when Braithwaite meets Denham in a boxing match and beats him with a swift jab to the solar plexus that winds him. What I like about this scene is the way that it is so central to Braithwaite's relationship with the class and the respect he is able to gain from them. Having triumphed over Denham in the boxing match, Braithwaite discovers to his delight that his class's opinion of him has greatly changed and he is looked upon with new respect, even by Denham himself. Note how Braithwaite comments upon the change in the class, talking about the rest of the physical education class:
When [Denham] was comfortable I continued with the P.T. lesson, which went without a hitch; the boys were eager to do their best, and went through the avrious movements without recourse to my prompting or direction; they now looked at me as if I had suddenly and satisfactorily grown up before their very eyes.
In particular, this change is indicated when Denham, without any prompting, address Braithwaite as "Sir," but sincerely, without any mockery. I suppose I like this scene so much because it shows how respect with a class is something that must be earned rather than just given as a free gift. Whilst I don't think I will be boxing with any of my "challenging" students, this scene shows the importance of establishing a rapport and gaining the respect of your class.