The Season 9 finale where Simon gave his goodbye speech after contestants from the entire run (seven of the first eight winners — David Cook couldn't make it — and 18 former contestants) sang together in one big song.
If we listed all the heartwarming personal stories shared by contestants, we'd be here all day.
On the season 11 Top 8 results show, contestant Joshua Ledet was feeling ill, but Ryan Seacrest showed that he genuinely cares for the cast by making it clear to Joshua that he should speak up if he starts to feel worse. He even told Joshua that he didn't have to come on stage to hear his result if he was not feeling up to it (he did though, and he was ultimately safe).
Colton Dixon's Season 11 audition had two of these. First, the judges forcing him to audition(he planned to just be there to support his sister Schyler) and Jennifer Lopez' speech to him on not giving up. The second came while Schyler, who was kinda robbed of her moment by the judges making Colton audition, was singing. The camera zoomed in on Colton, who was beaming with pride in his sister.
Colton got another one of these moments in Hollywood, when Schyler was eliminated. Colton, who usually was the Nice Guy, was angry and in tears, and told the cameraman that "This isn't right, she deserves to be here". Tear Jerker as well, but pure love between siblings as well
In season eight, Paula Abdul choreographed the disco week group number, and Ryan spent a good minute before and after the contestants performed talking about how wonderful a choreographer she was, and after, Ryan pulled Paula up to the stage and the contestants all gave her a bouquet of flowers nearly as big as she was, because she did so well. (Paula, understandably, was near tears.)
Simon Cowell actually apologised to Ellen DeGeneres for announcing that the ninth season would be his last at almost exactly the same time she was named to the judging panel (thereby giving off the impression that he was leaving because of her), even though the timing really was pure coincidence.
From season 6, overlapping with tear jerker; Sherman Pore, at age 36, was past the age limit of 28 for potential contestants, but he traveled to Los Angeles hoping for nothing more than to sing in front of the judges in memory of his recently deceased wife, who was a huge fan of the show. Naturally, the judges accommodated him, and he proceeded to give a heartfelt rendition of "You Belong To Me."