The trailer. Curse you, Arcade Fire, and your ridiculously moving music!
The opening incident that pushes Max to leave - namely, his older sister's friends destroying his igloo while he's inside. When he emerges, he's a blubbering mess, looking first at his sister's boyfriend who'd done it, then his sister... and neither one apologizes, or even approaches him to reassure the young boy, despite both looking remorseful while leaving.
This then leads to Max destroying a gift he'd made for his sister in a fit of rage, wanting to exact some form of revenge for her emotional abandonment of him. As Cinefix puts it:
Both moments are crude and hard. Not because of what happens to Max physically, but because they pack the sting of a white-hot childhood memory seared into your mind. A place in your psyche where you are forever that distressed child. Because we all lived it, one way or another.
The "Did you know the sun's gonna die?" scene. It's basically two children trying their best to cope with the concept of death. The scene is especially heartbreaking for people who went through this sort of realization in their childhood: that everything is going to end one day, and there's nothing you can do about it.
Alexander in general. He gets largely ignored by everyone for most of the film (Max doesn't even remember his name after spending a fair amount of time on the island, and having learned everyone else's), and even when they do notice him, he tends to get treated less than decently. Even Douglas, who is pretty much the Chew Toy, gets in on picking on him a bit. It also givessome insight as to why he makes snide remarks to Max over the course of the story; Max's arrival means that everyone else will be even less likely to pay any attention to him, so now he has to try even harder to get them to. And it still doesn't work! It's not until he legitimately gets injured in the process and just gives up that someone (Max, ironically) reaches out to him, and even then, his dialogue implies that he still harbors misgivings about it.
The situation of Bernard, the bull, is really kinda sad if you think about it. He just lumbers around for the entire movie, watching the plot happen and everyone else interact, and no one ever even makes an effort to talk to him. During the Dirt Clod fight, he gets immediately allocated to the "Bad Guy"s team without anyone ever asking if he wanted to, and later on, we see him get hit continuously by dirt clods. His only response is to give a melancholic grumble and sit down. And when we finally hear him speak just before Max leaves the island, Michael Berry Jr.'s contribution of Vocal Dissonance is jarring; we would expect a low, gruff, guttural voice, but instead we get a soft, timid, surprisingly high-pitched one. Just try and say you didn't react to that.
Near the end, when Max leaves the island. It gets sadder when the monsters all start howling.
The lead in to this with Max finding that Carol destroyed his own beautiful model forest, then Carol leaving indifferently when Max announces that he's going home. Carol goes to the remains of his model to cry, and he finds a heart (with a 'C' for Carol in it) that Max left for him. Carol then snaps out of it and runs to see Max as he sails off, arriving too late for a hug, but just in time to howl at him, and Max howls back. The look of guilt and regret on his face is crushing.
"I'll eat you up, I love you so."
Max's adventures with the Wild Things are sensationalized versions of the stuff he does for attention, except in happier contexts.
Max's reunion with his mom at the end is equal parts heartwarming and this, in a Tears of Joy sense that practically every parental figure can relate to.
:Both moments are crude and hard. Not because of what happens to Max physically, but because they pack the sting of a white-hot childhood memory seared into your mind.