One of the things that BioWare has done to add some depth to the story is having NPC's send letters to you at some point after certain quests. Some of them are quite touching.
For example, as a Republic player on Tatooine, one of the regional missions is to break up a slaving ring and free the slaves. After doing so, the freed slaves send a thank-you letter along with roughly 500 credits that they pooled together, saying that seeing you mowing through the enemy to come to their rescue was the first time in a while that they'd dared to hope.
After doing the main quests on Tatooine (Republic side), a local named Galen Besk asks you to help a bit more (i.e. bonus quests). Despite just hanging around the entrance to the spaceport, he seems to know more about you than he should. When you finish the questline, you get a chance to ask him about how he knows.
Galen Besk: Some Jedi fight to save worlds. Some fight empires. Some put their lightsabers aside and help in smaller ways.
A retrospective one in the "Deceived" trailer: Master Zallow is shown to press a Twi'lek gunslinger against a stone pillar with the Force - only to be engaged by a visibly pissed-off Darth Malgus. We later find out in supplementary materials that the Twi'lek, Eleena Daru, was Malgus' wife. That enraged attack on Master Zallow was essentially a successful attempt by Malgus to save the woman he loved.
Any Romance arc for a companion, that first kiss and all the others after it are all heart-warmingly sweet, and shows how much the characters truly love each other.
At the end of the "Hope" trailer, Jace Malcom sombrely stares at his many fallen comrades, Satele standing behind him. His narration states that even with great losses, "we are not alone." He fires a single flare into the air, which are followed by many more throughout the forest. From space, flare light dot the planet, in full view of the incoming Republic fleet. "Not alone" indeed.
One of the Republic quests in Corellia's Axial Park has you killing Mandalorians. Standard stuff, except the quest is called "Morale Boost" because the Mandos disabled a series of holostatues when they set up shop. The bonus part of the quest requires turning the statues back on and "give the people their heroes back". The statues are of the crew from the first KotOR game, including Revan. Three hundred years later, and people still remember them.
A Trooper player character gets an extra dose of warm fuzzies when the questgiver (a Republic soldier) starts feeling better immediately when he learns Havoc Squad is in the area. Turns out your squad has become heroes to the Force-blind Republic troops, because you prove that you don't have to be a Jedi to make a difference.
One with a side order of Tear Jerker: Running the Black Talon, your Imperial crew is trying to hunt down a defector. There's this Padawan that can't be older than about 16, trying to get the injured old man to safety. When the party closes in, the kid makes a futile Last Stand against your party just to try and buy the old man a little extra time. Even from the short exchange, you can tell she and the old man are fond of one another, enough to give her life for him without a shred of regret.
Generally, most companions have at least one special moment (unrelated to their Character Arc) where you can publicly express how much you value them. If that companion is with you at the time, picking that dialogue option leads to a massive Affection gain and a massive d'awwwwww.
Jedi Knight Storyline
On Nar Shaddaa, convincing Agent Galen to go on living even after he was forcibly turned into a Power Guard. It gets even better when he helps you bring down Lord Sadic.
After Kira's past as a Sith comes to light, most Jedi tell her that it's not what she was, but what she chooses to make herself that matters. For example, her Jedi master:
Kira: You need to know something, and I'd rather you hear it from me. I never told you about my childhood. I'm not who you think I am... [Cut while she explains] Bela Kiwiiks: So much about you is now clear to me, Kira, but you're wrong about one thing. You are exactly who I think: a strong, capable young woman. One day, you'll be a great Jedi.
Doc, who has a pissed-off ex-girlfriend in every port, offhandedly remarks that he's been elbow-deep in medical journals, and discovered the profoundly unnatural factoid that Wookiees spend their multi-hundred-year lifespans mated to a single partner. Later, if you romance him, his hilariously-awkward marriage proposal includes the line "I want us to spend the rest of our lives together, like the Wookiees."
Seeing Praven and Bengel Morr as devoted followers of the Light, if you have chosen to spare their lives.
Redeeming Tol Braga.
Jedi Consular Storyline
If you bring Lt. Iresso with you into the Healer's Temple on Voss during the main quest line, he'll offer to take your place and let the Mystics siphon his energy instead. Cue him collapsing in pain, but responding to to your voice; if you're romancing him, his sacrifice and your support is particularly sweet.
Nadia Grell's reaction if a male Jedi Consular proposes marriage to her is her jumping and kissing him in what is very similar to The Glomp.
Tharan Cedrax's decision to sacrifice his chance at eternal scientific fame to upgrade Holiday. Especially when Holiday starts worrying about how the upgrade might change her and make her stop loving him, and Tharan just offers gentle reassurance that he'll always be there for her, whether she loves him or not.
The "Light Side" ending of Act 1. Instead of killing Lord Vivicar, you can use your "shielding technique" to de-possess Parkanas Tark and instantly redeem the Jedi Masters' previous failure to save him on Malachor Three. Usually Dark Lords have to be killed; this is a refreshing and heartwarming change.
After completing the first story objective on Corellia, you're contacted by the First Son, who proceeds to monologue at you. You may choose to try to get Master Syo to fight from the inside. If Tharan is with you he will approve of any effort to help his friend, no matter how much he would normally roll his eyes at Jedi mysticism. It's especially heartwarming if you try to remind Syo of his friends, and namedrop Satele, Jaeric and Tharan, at which point the man himself weighs in:
It gets even better if you take Tharan with you on the last fight and choose the Light Side options. Syo is badly wounded and crushingly ashamed of what he has done under the First Son's control, but you and Tharan both reassure him that none of this was his fault, and you are still his friends, no matter what.
Another one with some Fridge Brilliance and another side of Tear Jerker: Senator Grell's will is read, and there is a last-minute addition; the Last Request to have your Consular train Nadia. Seeing as this guy was so determined to get help for his Force-Sensitive daughter that he was willing to dedicate his life to getting his people and their allies into the Republic, the fact he trusts you to take care of his daughter is humbling indeed.
Fuse. You just gotta feel for the poor guy, traitor as he was. He followed his squadmates for what presumably was a case of the politicians throwing them out the window just because they were a political threat, then on Tatooine he discovers the Imperials were using his expertise to murder innocents in order to field test the bombs that he designed, and by the end of the arc he's asking you to forget about his own life and get the son of a bitch who turned his gear towards it. Saving him and the message he sends later... just touched this Trooper.
For a male trooper, if you pursue a romantic relationship with Elara, she'll tell you that you are the only person that supported and treated her with any kind of respect since she defected to the Republic. The two of you will end up getting officially married and talk about one day having children together.
If you defeat M1-4X on Nar Shaddaa with Elara in your party, after the battle, a Republic troop will meet up with you and ask if you need healing. One of the answers is basically "Nah, my Combat Medic keeps me patched up." Cue massive affection gain from a flustered Elara.
The Smuggler's story is very funny and full of adventure, but one thing that truly shines is how personal the story is. The story is about the Smuggler, sure, you get to make a lot of heroic things, but in the end it still is a story about a freighter captain and his crew. That's what's most interesting, the whole crew becomes your family, and the game makes you really attached to Corso, Risha, Akaavi, Bowdaar and Guss, which in turn, each have developed loyalty and caring for you, the rest of the crew and the ship. It's a really heartwarming seeing how close knit the characters become.
On Nar Shaddaa, you repeatedly have the option to insult Drooga the Hutt and stand up for Bowdaar's mistreatment at his hands, even though it could risk your business deal with Drooga by doing so. The Smuggler can do so anyway, because it's the right thing to do.
While The Reveal of the class quest on Hoth is a Tear Jerker of the highest order, choosing to go down one path lands the resolution squarely here, if you convince Alilia to take Trick to Tython. You'll get a letter telling you that they're both okay—the Jedi are helping Trick, and Alilia has a job helping others like him.
Risha's friendship moment with the Captain, where she says that she would be willing to build a wing in her castle just for the captain visit her, perhaps with even a port where the Smuggler could dock the ship.
In-universe, performing many of the optional quests as a light-sider qualify as this across classes, but especially the Smuggler. Throughout the Prologue, you're in a race against time to catch up to Skavak and reclaim your ship, then in Chapter 1 you're racing to beat him to Nok Drayen's treasure, and throughout both of these and Chapter 2 you're trying to stay one step ahead of Rogun the Butcher's Bounty Hunters, who have no qualms about killing you. But that does not stop you from helping out others, from assisting in repairing Coruscant to rescuing a little girl's big brother.
A good example of this is in Oradam Village on Ord Mantell. There's a man whose daughter has been taken by Republic/Ord Mantellian soldiers, and as such no one will help him for fear of reprisal. The poor man is therefore reduced to begging help from you, a complete stranger that he's only just met. Yet you agree to help him, and a few hours later his daughter is back safe and sound. His wanting everyone to know who you are is understandable.
Sith Warrior Storyline
Commander Rylon's whole scene. First, you kill his captain, who is completely loyal to Rylon, then in his final moments discovers that his commander is actually an Imperial Spy. Then, Rylon starts to talk about how he already knew that Baras was going to kill him eventually, and that he would face his death with honor. The last thing he asks of you before the fight is if his son, who you met earlier faced his destiny (that is decided by you) with his chin up.
The Light Side route is actually rather heartwarming. While you still are reluctantly forced to end his life, you can admit that you have come to greatly admire him, offer him to grant his last request, tell him that you spared his son's life (or, alternately, if you did kill his son, lie about how he faced his end bravely), before finally letting him to die a soldier's death, with his honor and dignity intact.
Proposing to Vette before she has a chance to propose to you has her literallynote (The actual definition of literally, not the "for emphasis" definition) jumping with joy. It's so heartwarmingly cute.
Vette's reunion with her old gang on Nar Shaddaa following the "Star of Ryloth" mission.
Male Warriors who turn down the flirt option with Taunt get the option to hint at their growing attraction towards Vette, leading to an amusing exchange between the Warrior and Taunt which Vette remains completely clueless about.
Warrior: I've already got all the beautiful Twi'lek women I need, thanks.
Taunt: Oh, you're... I see. *Laughs* Well, good luck cracking that safe!
Vette: *Oblivious* Am I missing something?
The Light Side route on Tatooine has the Warrior later receive a message from Sharack Breev, who was inspired by your example to let go of her fear and venture beyond the Forbidden Pass. She eventually came across the home of Master Yonlach. Instead of finding the slaughter she'd expected, she was pleasantly surprised to find the Jedi Master alive and well. They even ended up discussing his bizarre encounter with an unusually kind Sith.
Vette is perplexed at why you chose her to be your significant other, pointing out that you're a Sith lord who can get anything you want and she's just a former slave. The Warrior's flirt response leaves her speechless.
Warrior: You're beautiful, charming, sometimes crazy, and always exciting. What man wouldn't choose you?
Sith Inquisitor Storyline
After Lord Zash takes over Khem's body when he interrupts her ritual at the end of chapter 1, among your dialogue options are "Khem Val? Are you still there, my most morose monster?" and "Go away and give me Khem back." (Khem approves +120) Particularly heartwarming in that the Inquisitor is not all that keen on referring to Khem by his name in dialogue.
If you pick the right dialogue option when facing Overseer Harkun on Korriban after passing your trials and being made an apprentice, you will genuinely thank him, pointing out that the adversity and struggle the Overseer put in your way allowed you to grow strong and taught you a great deal. After all, strength comes from adversity, and learning strength is the Sith way. Harkun, a racist and classist jerk of the first order, is so shocked and touched that he actually calls you "apprentice" instead of "slave" for the first time, and wishes you good luck, in his own gruff way.
Convincing Andronikos to let go of his desire for revenge against the man who lead the mutiny against him, who until then he'd been obsessively pursuing across several planets, all by uttering four simple words;
When the Inquisitor is given a ship, one of the dialogue options is graciously thanking Zash for it. When you get to the spaceport, the Inquisitor runs over to see the ship and the music is bombastic and happy. For the former slave, this ship represents the freedom you now have and even though you're still following Zash's orders, you can travel the galaxy without being beholden to anyone.
Ashara and the Light Side Inquisitor have several of these moments, as instead of attempting to corrupt her to the Dark Side, they instead become a mentor figure who enlightens her to a larger view of the Force, which encompasses both Light and Dark, something the Jedi often refuse to acknowledge. They even have the option to steer her away from becoming too much like a Sith and even ask her to be their Morality Chain and keep them on the straight and narrow.
Towards the end of chapter two, Zash will suggest that the Inquisitor takes an official apprentice from the Korriban Academy for the additional prestige it confers. One possible reaction is "Shove it, I already have an apprentice!" (Ashara approves +120)
Likewise, throughout her romance arc with a Male Light Side Inquisitor, they repeatedly respect her wishes to take their relationship slowly, has no traces of their usual snark and display more affection towards her than anyone else in the entire game.
Out of all the lines in the romance quest, this one is sure to warm the coldest of hearts.
Ashara: If you hadn't become Sith, what did you want to be?
Heck, it's heartwarming if you are Dark Side and choose the right options. You're a complete and total Blood Knight who ruthlessly manipulates others and cares only for his own power...and his Togruta lover. The voice-acting when he speaks to Ashara is just fantastic, as his tone switches from cruel and mocking to something close to soft and tender.
Bounty Hunter Storyline
At one point early on in the story, if you're male, Crysta Markon will flirt with you and ask if your partner, Mako, is your little sister, and then say that she at least can't hold a real man's attention. One response is to say that Mako is the only person in the galaxy you care about, rendering Mako speechless.
After one has earned considerable affection from Mako, she can be pursued as a romance option and married. One such message from her reads "Did you know Czerka Corporation has wedding registries? Well, they do. It was kind of fun to set one up even though no one's going to get us anything on there. Well, almost no one. I saw the best set of shape-conforming blast armor on sale. I think it'll look good on you, so really it's a present for both of us. Love you, big guy."
On Hutta, one of your bounty runs is to capture a Republic scientist that your rival kidnapped in order to lure that rival into a trap. Well, you manage to nab your quarry and lie in ambush for the goon. After killing your rival to get the token you need to join the hunt, the scientist is scared out of her mind. she does have a price on her head, after all. Choosing to let her go, Mako hands the lady a holdout blaster and gives her directions on the safest way to the spaceport. Seeing as you and Mako barely have two credits to rub together at this point, it's an Honor Before Reason gesture, but the scientist sends back a letter of gratitude with some credits, relieved that there is some shred of decency left in a Wretched Hive like Hutta. This Hunter has been a light-sided, closeted Republic supporter since.
This mission also brings out an option for Mako to be your Morality Pet. You can contemplate aloud the fact that the scientist still has a bounty on her. Mako will protest...and you can say that you won't do anything that makes her uncomfortable, and let the scientist go. This option actually gets you more approval from Mako than just letting her go outright.
When facing Reneget Vause in an honourable Duel to the Death on Hoth, the Light Side route has the Hunter make one of the rules that if they should fall in battle, their companion gets to walk away without harm. Considering that the Trandoshan is threatening to eat them if they lose, it's rather touching that the Hunter is more worried about Vause harming their friends than their own safety.
On Hoth, not only can you rescue Blizz from certain execution, but the Light Side option lets you convince the Admiral to have him pardoned of all crimes against the Chiss Ascendency.
And if you complete all of his missions, Blizz simply says...
"Blizz think Blizz home."
When playing as a female bounty hunter, Torian can be pursued as your love interest. Right before he ask you hand in marriage near the conclusion of the romance side quest, he will say 'I love you' in Mando'a, and you have the option to respond by tell him the same thing, also using Mando'a. This will gain you a massive boost in affection with him. Watch here.
The scene plays out differently if the bounty hunter refused the offer to become a Mandalorian. Torian instead asks two questions: the first, if she'd raise their children Mandalorian. And if yes, what would she do, purely hypothetically of course, if he was to ask her to marry him. Just as sweet.
Imperial Agent Storyline
After gaining enough affection with Kaliyo, she gives you a mission to help her bail a friend out of jail. It turns out that this friend has a bounty on her head that Kaliyo wants to collect on, all due to a perceived slight. If you opt for the LS option and let the girl go, Kaliyo is pissed at you. The only option that doesn't result in a loss of affection is telling her that "You're better than this, Kaliyo." Given she's an Ax-Crazy anarchist who hates authority, the fact that this gets through to her is oddly touching.
When Kaliyo suggests marriage in the only way she can, the agent can agree on the condition that Kaliyo finally admits that she loves him. Cue Kaliyo getting almost adorably flustered before saying she does. Her reassuring return to form immediately afterwards by threatening to do harm unto the agent's person should he tell anyone only adds to the strangely heartwarming nature of the scene.
The romance storyline with Vector Hyllus has several such moments. At one point, during a 'private moment' with the female agent, Vector represses the bond to his hive so that they can get some true alone time. Possibly made even more heartwarming if the agent chooses the "This isn't you" response, and accepts him as a Joiner.
Later on, the female agent can suggest becoming a Joiner as well. Vector's reponse is beyond sweet.
Vector: It would mean giving up everything you are, everything you wish to become. We have no desire to see you do that. We love you. Will you marry us?
Vector really does have quite a few heartwarming lines in the romance storyline, in his own Killik-y way:
Vector: There are patterns in your electric aura that are yours alone. They taste like frost and spices. We will remember them always.
Vector gets a lot of these, period. There's his friendship with Lokin, which has the two of them being so reassuringly normal, which is a bit of a feat considering their respective natures. Nevertheless, they plan hiking trips and visits to the opera. At the end of their dialogue based mini-arc, Vector explains the nature of their friendship in a way that resonates with the themes of the Agent's story as a whole: No, he doesn't trust Lokin. Of course, the good doctor always has an angle. But you don't have to trust someone to like them and be their friend.
One more for the Killik that requires a bit of knowledge of the setting. See Vector's assurance that "We will remember them always." above? That's literal truth: the nest remembers everything that has ever happened to any of its members. By the time Alderaan was blown up in ANH, the Killik were a distant legend on the planet—they had left a long time ago. So assuming the descendants of Vector's nest are still around somewhere in the galaxy, there really are people who can remember the Agent and their crew, three thousand years later.
If you play a Chiss, after completing the Hoth storyline, Aristocra Saganu contacts the agent to declare him or her a merit adoptive of his family. A player familiar with the species will recognize this as a major honor allowing exceptional Chiss to join ruling families, one that will later be given to Grand Admiral Thrawn.