After Phil Stacey was eliminated, he sang what was probably the most appropriate song imaginable...Blaze of Glory by Bon Jovi. He belted it out, and at the end embraces his fellow contestants. http://www.youtube.com/watch?v=oIjmxdUkbf8
Kris Allen. "Heartless". His performance is so awesome that Randy Jackson declared it "better than the original and better than the Fray's version." It is so awesome that some consider it the only reason he made Top 2 and the main reason he won the entire competition.
In season 5, the top 4 was made up of early favorites Taylor Hicks, Chris Daughtry, and Katharine Mc Phee, as well as Elliott Yamin, who had not received much, if any, screentime before making the top 24. Despite being widely regarded as a fantastic singer, he was predicted to be the contestant to leave once the final 4 were announced due to him being less popular than the other three. But not only did Elliott sing well enough to escape elimination, he escaped the bottom 2 entirely while Kat and Daughtry wound up in danger (I think we all remember how that ended...); the fact that a contestant received more votes than two early favorites (possibly three; he could have easily beat out Taylor) due solely to his two fight-back performances that night is a fantastic achievement. Elliott went home the following week, but it's hard to deny him the awesomeness of his acheivement of outlasting the best-selling male AI contestant ever.
Going on to outsell the two contestants who beat him is pretty awesome too.
In Season 10, the judges saved Casey Abrams by cutting him off almost as soon as he started to sing, which made him think he was getting eliminated, and then telling him he was saved, which almost literally shocked him.
In Season 11, they saved Jessica Sanchez by climbing up on the stage while she was singing, commandeering the microphone, actively putting her on the seat, and then ranting to America about how they're supposed to vote for the best. Even Ryan Seacrest said that the judges "did the right thing."
if memory serves, Randy also called out at least one of his performances as better than the original.
Carrie Underwood's performance of Heart's "Alone", which made Simon predict that not only would she win the show, she would outshine any other previous Idol winner. As of 2012, Carrie has not only outshone the previous Idol winners, but all subsequent ones as well.
Ryan Seacrest, during the top 7 results of season 7, split six of the seven contestants into two groups and said that one was safe, and one was not, and that David Archuleta, who was the only one explicitly stated as "safe", had to guess which one was the safe group and join them. Archuleta's response? Was to sit down in the middle of the stage and not move, despite Ryan's multiple attempts to get him to move or get up.
The very first season, Kelly Clarkson emerging from the pack with her rendition of "Natural Woman" and suddenly perfroming a whistle note that brings the audience to their feet.
Paula Abdul, most well known on the show for acting "whacked out" (a side effect of being in chronic pain for a decade and thus on painkillers), choreographed the group number in season eight's disco week. After years of seeing her act that way, seeing her show the choreography to the Idol contestants and being in her zone, completely serious and, to use her words, acting like a drill sergeant, it's a serious moment of awesome.
Allison Iraheta, in her very first live performance on the show, sang Heart's "Alone", and promptly blew everyone else out of the water, even Kris Allen and Adam Lambert.