A lot of people comment on how their family members, mainly their mothers have played Animal Crossing with them which is naturally heartwarming but then they say they've passed away. You could even argue that the fanfic Slice of Heaven was made with this concept in mind (unless the author can actually relate to it).
In Wild World, sometimes when visiting the cafe, you'll see Rover from the first game sitting at the counter. Hearing him fondly reminisce about traveling the world and meeting so many faces, and how he misses those days. It just sort of gave me a pang of nostalgia.
Any time a villager you've really grown to like moves out. Doubly so if you did try to convince the villager to stay but failed for one reason or another, or didn't realize that said character was leaving until it was too late thanks to none of the villagers telling you (the latter was what happened to Ai in the anime movie). It really hits you when continuing your town file that a character finally tells you your favorite villager is one day away from moving.
Even worse if it's the first Animal Crossing, where you cannot get their portraits to remember them.
The villagers' panicked reactions when you consider deleting a town, and their resignation should you choose to.
Try playing the game years after you've long since forgotten about it. The villagers all remember how long it has been since they last saw you. Watch the tears fly. In New Leaf, Isabelle will note that she tried her best to keep the town running while you were away for a year.
Even worse, some say the number of weeks instead of months. Their reason for this is because the larger week number makes it look more significant than the smaller month number. They're right.
When you become friends with Sable, she'll tell you at one point of how she had to give up her normal life to take care of Mabel, due to their parents' early death.
One of the stories Sable tells you is of how she and Tom Nook used to be best friends in their youth. But after leaving for the big city as a Wide-Eyed Idealist and promptly being broken there, Tom returned as a different person, and their friendship never really recovered. Sometimes both Sable and Tom will express sadness about it. Thankfully, New Leaf at least hints that they're on the road to rebuilding it.
Another is that they had a third sister, Label. Not long after the tragic accident, Label and Sable had a heated argument, which resulted in Label leaving town to become a fashion designer. During the events of City Folk, she tries to mail her sisters to try and put the pieces together, which causes Mabel to get depressed about the situation. The player's help can soften things up, but even in the end, Mabel wishes her sisters would get along again. Fortunately, they do by the time New Leaf kicks in, as evidenced by Label working at the same shop as her sisters.
Tom Nook, the guy everyone likes to portray as an evil mafia boss who traps the player in indentured servitude, becomes a lot more sympathetic once you hear his backstory. (The game getting rid of that mechanic entirely by New Leaf probably doesn't hurt, mind you.)
This sudden little stab in the heart from Lyle in City Folk. He used to play in the snow with his daughters, but stopped, with the implication being that he abandoned them.
Or worse, they abandoned him, have grown up and moved out.
Deny a Cranky villager his medicine in Wild World when he's sick, and suddenly that animal who's normally a Jerkass begins having flashbacks and crying for his mother. It's absolutely heartbreaking.
Some of Kappn's shanties. Not a dry eye in the room when he sings some surprisingly deep, inspirational songs.
Don't cry about dreams that don't come true today. Sometimes ya needs ta try another way!
Whene'er I'm sad, that's when me wife'll say, "The sun will rise and warm you today!"
Oh, this ain't good-bye. I will see you again. That's how it goes whene'er yer my friend.
Sometimes your heart tells you when you're in love. Your tummy too. Both can sure make you sick.
One time I cried till no more tears would flow. When your heart's broke, that's just how you're gonna know!
In the film, Sally's letter admitting she didn't tell Ai she was moving away, as she knew it'd make both of them cry.
The film also implies something of a romantic nature happened between Whitney and Apollo, and it didn't end well.
Aika Village, arguably the most famous dream address in New Leaf, is an equal mix of horror and this. There are many theories behind the true story behind it, but no matter the exact structure, it typically revolves around a young girl whose life has somehow gone horribly, horribly wrong. The town is laid out so that a fixed path of travel is mandatory; the first area is beautifully tended and cheerful, with the titular girl saying "I love my mommy" when spoken to. However, as the path continues, the town starts getting more torn-up, the houses start getting more unsettling, and Aika's speech, while saying the same thing, begins getting steadily more garbled. Just how little information you can get from a town leaves you stunned and wondering what on earth happened to this girl. The dream address is 2600-0218-7298.
New Leaf's 1:00 AM music can make a completely innocent theme into this.
New Leaf's 4:00 AM theme can easily become this, as well. It doesn't help that it plays in the the above-mentioned Aika Village.
From the original GameCube version, there are the the 7:00 PM theme, the 8:00 PM theme, and the 8:00 AM theme, which evoke odd intermingling senses of nostalgia, sadness or wistfulness, and acceptance/peace all at once.
2 AM in Wild World and City Folk/Let's Go to the City, in contrast to its fast-paced Super Smash Bros.. Brawl remix and the funky 2 AM theme from the GameCube game, is one of the more emotional songs in those two games, along with other late-night songs such as 11 PM, 12 AM, 1 AM, and 5 AM. (4 AM is very minimalistic and 3 AM is strangely upbeat, though to a lesser extent than 2 AM and 4 AM from the GameCube game).
The fortune for the Arwing from New Leaf.
Your friends need you. They always need you. They will never stop.
When you accidentally hit a villager with a net. Nothing is more scary then the fear they hate you.
Refusing to play supermarket with Kapp'n's daughter Leila. Making a little girl cry can be a real punch to the gut for some players, and you will feel like a "big meanie".
Though most players may think him as a total Jerk Ass in previous games there's the fact that Mr. Resetti lost his job in New Leaf despite him still being perfectly capable of performing it. All because the Reset Surveillance Center was shut down, it may make certain players see him in a whole new light and feel sorry for him and realize that all his brother Don's claims of him doing it because he cares were true and also realize that it's the only job he's capable of (considering while other special NPCs have taken new jobs or kept their old ones, or in Tortimer's case have retired, notice how Resetti is the only one who's lost his line of work). Thankfully you can use your mayoral powers to restore the Reset Surveillance Center and give him his job back.
If you visit Phyllis at The Roost in New Leaf, she'll reveal to the player that she didn't really enjoy working under Tortimer because all he did was sleep and didn't leave the town hall unless there was an event. She goes on to admit that if you'd been the mayor back then, she would've enjoyed her job more. It really drives home the fact that Phyllis is just crabby because she's so overworked.
Especially the fact she only works nights, the sad lonely time when most players are asleep. It's hard to appreciate your job when your more popular sister gets all the attention.
The letter that comes with the townspeople's Time Capsules often starts with: "To a better me..."
In the original game, after you leave the New Year party the music is a slow, soft version of "Auld Lang Syne."