Quite simply the ending: Red meets Andy on that Mexican beach, both of them free as a bird and as happy as can be to see each other again. The smiles they exchange and the way they run up to each other and hug says it all, by holding on to hope that they would be free one day, their spirits finally triumphed over all obstacles.
Shit, I'm getting misty-eyed just thinking about that.
Red: I find I'm so excited, I can barely sit still or hold a thought in my head. I think it's the excitement only a free man can feel, a free man at the start of a long journey whose conclusion is uncertain. I hope I can make it across the border. I hope to see my friend and shake his hand. I hope the Pacific is as blue as it has been in my dreams. I hope.
The look on the convicts' faces as they hear the beautiful strains from Le Nozze di Figaro play over the loudspeaker. This is probably the first time any of them have heard music in quite a long time, and the overhead shot of them staring at the loudspeaker in unbridled awe never fails to bring tears to this troper's eyes.
The scene at the end when Red finds the package Andy left for him. After a whole movie of having believed that hope was "A dangerous thing", Andy has proven him wrong and has now given him something to hope for as well. The real CMOH of this scene however is sealed with the closing portion of the letter:
Andy: "Remember Red, hope is a good thing, maybe the best of things, and no good thing ever dies. I will be hoping that this letter finds you, and finds you well. Your friend, Andy. "
Tommy's Undying Loyalty towards Andy. When Norton question him on Andy's innocence, Tommy repeatedly states that he would be willing to stand before a judge and swear on the Bible if it meant Andy could walk out of court a free man. Norton murders him in response, as he figures he cannot allow Andy to walk free - ever.
All of Andy's attempts to help the prisoners make something out of their lives in prison with his library, and his helping them prepare for after they get out. Given the heartbreaking fate of Brooks this becomes all the more poignant.