1. When Homer quit his job for Maggie.

"And Maggie Makes Three" (Season 6, Episode 13)
Homer tells the story of how he quit working at the nuclear power plant and took up his dream job at Springfield's bowling alley. But when he found out Marge was pregnant with Maggie, he begged Mr. Burns for his job back and accepted the humiliation and sacrifice to his own happiness that came with it. We find out that the reason there aren't any pictures of Maggie in the family album is because they were used to decorate the motivation he creates for himself: "Do it for her."
2. When Mr Bergstrom left Springfield, and Lisa, behind.

"Lisa's Substitute" (Season 2, Episode 19)
Mr Bergstrom is Lisa's favourite substitute teacher, but then he has to leave. Lisa is distraught, but he hands her a note before he's gone forever. His parting advice is simple and effective. Whenever she feels "all alone" and like "there's no one to rely on", she should remember what's written on it.
3. When Maggie was tempted by the Flanders family but still picked the Simpsons.

"Home Sweet Homediddly-Dum-Doodily" (Season 7, Episode 3)
After staying with the Flanders for a little while, Maggie goes to make a choice between them and her actual family. It's looking a bit rocky but Marge steps out of the bushes at the last minute and Maggie naturally crawls toward her. Seeing the two reach out for one another is a genuinely lovely moment.
4. When Lisa thought she was losing her intelligence.

"Lisa the Simpson" (Season 9, Episode 17)
Grampa Simpson tells Lisa that she is destined to endure the "Simpson Curse" of a failing IQ. She thinks her inability to finish a puzzle is proof of that. Lisa says goodbye to her intelligence on TV. She asks everyone to hold on to their smarts, and to look for beauty, whether that be "in the swirl of a galaxy, or in the swirl at the centre of a sunflower".
5. When Mr Burns was reunited with Bobo.

"Rosebud" (Season 5, Episode 4)
Mr Burns' reunion with one of the few things that he genuinely likes, his teddy bear Bobo, is quite sweet. "I'll never leave you behind again," Mr Burns says, in his wafer-thin voice. It's a moment of surprising loyalty, and an interesting change from his usual ruthless tendencies.
6. When Nelson got attached to Marge.

"Sleeping With the Enemy" (Season 16, Episode 3)
When Marge decides Bart can be undeserving of her motherly love, she extends her hand to Nelson. We see Nelson doubt the kindness being offered to him, and then take full advantage of it. The scene reminds you that he might be a bully but he's also just a kid.
7. When Bart sold his soul to Milhouse.

"Bart Sells His Soul" (Season 7, Episode 4)
After selling his soul to Milhouse, Bart realises the importance of what he's done. He has a dream where he watches other people play with their souls while he goes without. Even though he gets it back in the end, it's unsettling to see him struggle and learn that being soulless isn't a very good thing at all.
8. When Marge reassured Homer of her love for him.

"Simpson and Delilah" (Season 2, Episode 2)
Homer grows a full head of hair thanks to some special shampoo, but it all falls out again. He asks Marge why she loves him when he's "bald and fat". Marge holds him close and sings "You Are So Beautiful" by Joe Cocker to him.
9. When Bart got caught shoplifting.

"Marge Be Not Proud" (Season 7, Episode 11)
Bart disappoints Marge when he's caught shoplifting. He tries to fix his mistake, and it all turns out well in the end, but Bart's fall from grace in Marge's eyes makes for some sad Simpsons viewing.
10. When Lisa dumped Ralph on TV.

"I Love Lisa" (Season 4, Episode 15)
"Watch this, Lise," Bart says. "You can actually pinpoint the second when his heart rips in half." It's Ralph's complete harmlessness when Lisa dumps him on TV that makes this moment so painful to watch.
11. When Apu stopped owning the Kwik-E-Mart.

"Homer and Apu" (Season 5, Episode 13)
Apu's song turned soliloquy "Who Needs the Kwik-E-Mart?" is all fun and games until he realises the answer to the question. Who needs the Kwik-E-Mart? Apu does.
12. When Marge realised she'd changed for the worse.

"Scenes From the Class Struggle in Springfield" (Season 7, Episode 14)
Marge picks up a Chanel suit in a discount store and is invited to Springfield Country Club, but the pressure to keep up with Springfield's richest interferes with the happiness of her family. She realises how much she's changed when Homer says, "You kids should thank your mother. Now that she's a better person, we can see how awful we really are." And then the Simpsons skip the country club to go to Krusty Burger.
13. When Homer got a glimpse into Lisa's world.

"HOMЯ" (Season 12, Episode 9)
Homer discovers the reason for his low intelligence: a crayon lodged inside his brain. The most touching part of this episode is his realisation that being smart doesn't necessarily mean being happy. He asks for the crayon to be reinserted, and leaves Lisa a note: "I'm taking the coward's way out. But before I do, I just wanted to say being smart made me appreciate just how amazing you really are."
14. When we found out Principal Skinner isn't actually Principal Skinner.

"The Principal and the Pauper" (Season 9, Episode 2)
After finding out that Principal Skinner isn't actually Principal Skinner (and the real one is thought to be dead), the relationship between Skinner and Agnes gets more interesting. She isn't his mother. But he withstands her insults and weird affection because she needed someone to be her son and he decided that someone was going to be him.
15. When Bart got an F.

"Bart Gets an F" (Season 6, Episode 1)
In an episode that shows Bart at one of his lowest points, we see him failing fourth grade and trying to stop it from happening. The emotional heart of the episode is seeing him still not make the grade – and this time he actually tried.
16. When Homer's mother came back and then left again.

"Mother Simpson" (Season 7, Episode 8)
Homer discovers that his mother Mona faked her death. He has her in his life for a little while, but then it turns out she has to leave again. "Don't forget me," he says as he waves her off into the sunset. She tells him he'll always be a part of her before she is driven away. And then Homer is motherless again. He sits on his car, staring into the distance where she once was. It's a moment that raises lumps in throats every time.
CORRECTION
The original Skinner is not canonically dead. He is simply another person. An earlier version of this post misidentified this.