Or more specifically, Aang.
While he isn't as reviled as Ash Ketchum or Shinji Ikari in terms of male protagonists people hate, he still gets a lot of hate thrown at him, especially after the grand finale. His finishing move was a Grand Ass Pull, his angst, the fact that he gets the girl.
1. "He's too whiny and angsty!"
So for the first two seasons Aang's a plucky kid hero who cries or gets pissed off now and then, but for the most part shakes it off with a smile. Then in the season three opener, he's a bucket of angst. I can see how that might be a little jarring, but let's be realistic here: He died, came back from the dead, and then fell asleep long enough to grow hair without realizing it. When he woke up, Ba Sing Se was gone and everyone was telling him to just fake the dead for a while. Aang wasn't gonna be happy about that for obvious reasons. :P But he accepted it eventually.
And in "Ember Island Players" his angst and issues with the play might've been a little over the top (it upset him so much he would've gone into the Avatar State? Wow. XD), but this was right before the final battle. Also, Aang is a 12 year old boy so he's naturally gonna feel insecure about people laughing at the obviously female version of himself nancing and prancing on stage. :P And really, Toph was the only one actually enjoying the show until the downer ending, so...
But I digress. Point is, Aang's a 12 year old with a huge responsibility, he's been FROZEN for 100 years and didn't realize there was a war going on and that his people were wiped out. He has a lot to deal with, he's only human, of course he's going to break under the pressure sometimes.
2. "The Lion-Turtle and Spiritbending were an ass pull! He cheated! He was too pussy not to kill Ozai like his past lives said to! Aang was just being selfish and wanted his own way! It was just a cheap way to keep the precious hero from getting blood on his hands!"
Okay, I won't argue that energybending and the lion-turtle could've used more elaboration. Mostly because Bryke would've done an awesome job with it and it would be really cool to learn about. XD
But Aang not killing Ozai wasn't him being selfish or spineless, and had nothing to do with keeping the hero's hands clean. Aang has never liked the idea of killing because that's the way he always was. And energybending was Aang saving the world as Aang...and it worked. And even if he's alive Ozai won't be a threat anymore because he's been locked away in prison, and I don't see Zuko letting him out anytime soon. Or ever.
Aang WAS ready to take Ozai's life, even after he found the lion-turtle. He didn't understand its words until the last moment, when he decided he was gonna end Ozai his way. Avatar Kuruk said "shape your own destiny", and that's what Aang did.
3. "He doesn't deserve Katara, they just let him have her as a prize for saving the world! It's so cliche, come on, the hero should never get the girl, it's more realistic!"
This is one of the more ridiculous arguments against Aang and against the Aang/Katara pairing. Aang loved Katara, and Katara loved Aang. She didn't hook up with him just because he saved the world, and he didn't act like he "deserved her" for doing so. The pairing had been planned from the beginning, and just happened to become official at the very end. Saving the world and getting the girl are not mutually exclusive things.
Realism also has nothing to do with it. Avatar is an idealistic show, and frankly I think having it end with Katara rejecting Aang and Aang learning An Aesop that it's okay to be single would've truly fucked up the ending, not to mention the show's entire premise. Forced downer endings are just as stupid and unrealistic as forced happy endings.
Also, I've come to fucking hate "deserves/doesn't deserve" as an argument for or against pairings. Katara isn't some sort of prize to be won or lost, she's a person. A person who chose the boy who happened to be the hero on her own.
So to sum up: Aang is a human being who struggled with his destiny, chose to save the world his own way and happened to come together with the girl he loved in the end. NOT a selfish whiny loser who shouldn't get the girl because "it's more realistic".