Ever take a scary Pokemon down to meet Dr. Footstep? It can melt anybody's heart to learn that even big, scary Pokemon like Gyarados love them. In fact, said Pokemon only feel missunderstood.
"There are no Pokemon who dislike humans. Only humans who dislike Pokemon. We [Pokemon species] are especially shunned.. .but [name] always treats me as a friend and partner." This is the quote said by "dark" Pokemon.
How about when you find out a flippin' Legendary loves you? If you treat it well, this itself goes Up to Eleven with Arceus, the Pokemon equivalent of god.
Hearing Dr. Footstep's translation of your mons' innermost thoughts are adorable in general:
Most Pokémon: <Name> is... a remarkable human and Trainer. That I can always perform the best any <Pokémon species> possibly can... I attribute that entirely to my partner <name>. When we travel, I can see wild Pokémon eying us enviously...
Scary Pokémon: There are no Pokémon that dislike humans... Only humans that dislike Pokémon... We <Pokémon species> are especially shunned... But <name> always treats me as a friend and partner...
Slow Pokémon: Hnurrr... Me? Uhm... Other Pokémon, they say to me... "You're not thinking, are you?" How insulting they should say so. It isn't true what they say. Think, think, think, I do that. All the time, I think hard how to help <name>. I concentrate! Hnurrr...
Other Pokémon: If I said there can be friendship between Pokémon and people, will <Pokémon species>'s friends understand? But if you see <Name> and <Pokémon Nickname>, you will understand. Because <name> and <Pokémon Nickname> are friends!
There is another set for Pokémon without footprints, per all 5 levels. Plus another for voiceless Pokémon. That gives a grand total of 41 dialogs. All to be seen here
As well, some Pokemon will remark how it's too bad that their trainer is only a human.
Barry getting teary-eyed when his dad first is introduced in Platinum. Just something about the normally rather cocky Keet getting all emotional was too adorable.
Charon and Rotom's backstory. All he wants is a hug!
In Platinum, if you go home during Team Galactic's attacks on the Lakes, Mom comments that she sees the news and worries about you. Aw, Mom...
Looker's farewell at Stark Mountain in Platinum. Especially heartwarming as it was revealed in the Platinum Let's Play that Nob Osagawara, longtime freelance translator who translated the text of every main series game through Platinum and known as Doug Dinsdale on the Something Awful forums, used Looker's dialogue at the end to say goodbye to the series and to the fans. "Long may you cherish Pokemon!" Will do, Looker and Nob. Will do.
Your rival gets a small one in Platinum after you beat Volkner and then are about to get HM07 from Jasmine.
(rival): Hey! (player name)! What the... You went and got all eight Gym Badges of Sinnoh! Aww, you... That's wicked! I envy you! But don't think you've won! (turns back) Because I'm going to be the toughest Trainer ever! You watch me! Before I was dreaming. I just wanted to be like my dad. Now, things are different. I thought about what being a Trainer involves. I thought about what Pokemon are. We're going to get tougher one step at a time! (turns to you) I'm not going to challenge you to a battle now. I hate to say it, but I know you're tougher than me. But not a whole lot tougher. Just a little. Our difference is about a yard? Nah, not even an inch. ...What I'm trying to say is that I will catch up to you in no time. If you don't hurry to the Pokemon League, I'll leave you in my wake! (flees)
Coming from a character who talks about becoming the strongest trainer ever every ten seconds, he does gain some development here.
Another one in Platinum, when Cynthia thanks you for saving Giratina from the Distortion World, then Rowan and Dawn/Lucas do the same thing.
Play the game on your birthday. When you go to Sandgem Town and talk to Lucas/Dawn, he or she will wish you a happy birthday. Simple, but effectively touching.
On Route 210, right before you get to Celestic Town, in Platinum there's a Veteran who wants to retire and make you his final opponent. After you beat him, he feels a rush and says that he perishes the thought of retiring, realizing there are still strong Trainers out there. In other words, you managed to convince him to not retire by beating him. That's pretty nice.
Tower Tycoon Palmer is not just the leader of the Battle Tower, but your rival's father, which gives a nice personal connection to his battles. The greatest moment is if you defeat him a second time, which is no easy feat considering you have to survive through 48 other trainers with the odds against you. He reminisces about how he started off his journey by miraculously fending off a wild Pokemon with the player character's father (one of the only times he is mentioned, aside from Norman) and wonders how the Professor is doing. That memory and his speech about the world holding infinite promise for you makes the victory feel like a monumental accomplishment. He gives the infinite promise speech in HeartGold and SoulSilver as well, but it loses much of its impact without the family connection.