Awesome movie. Intense, greatly acted, greatly written, great music. Lot of other reviews touched on many things, but here is why it was so special for me:

1. The structure of the story. On normal story (look up Hollywood formula) you have
first: introduction with the main character taking a choice (without it you have no movie) e.g. Matrix taking the red pill
second: character tries to achieve his goal, but gets to a low point.
third: climax, character achieves his target emerging from the low point

this creates emotional satisfaction. Some movies play with this "formula". And it's the case of the Whiplash. Here, everything happens in last 5 minutes, which then creates the "Wow" you can read in reviews. The whole movie is introducing characters their motivations, and gets them to the base for the climax. (decision to go back, he is at the lowest low, and shines).
further why the last 5 minutes are so strong is my second reason:

2. The Antagonist (Simmons) - greatly acted, he as the main obstacle Teller is against. But Simmons has its own story arch that climaxes at the same time with Tellers, making the last 5 minutes two times stronger! (spoiler: wanted revenge, did not work out, wanted in even more, but then found what he was looking for: someone who will shine, that he helped = smile, satisfaction)

feel free to discuss if you don't agree :-)