Damon and Stefan had a less-than-brotherly relationship sometimes, but when it came to the things that really mattered, they stood for each other. Their mutual love for Elena largely dictated the direction of The Vampire Diaries, but the worst threats they faced were mostly because of their poor judgment and past mistakes and not Elena's fault.
Their past was full of interesting events including their entanglement with Katherine which caused them to become vampires and hurt each of them deeply. Despite all that pain and animosity, the brothers had a resilience that saw them get past their hardest moments and find common ground. Their story, although messy is one of love and tolerance as their own words prove.
"Your Answer Could Be Summed Up Into One ... Word: Elena."
"Pilot" (Season 1, Episode 1)
Damon wanted to destroy Stefan's relationship with Elena because he blamed him for everything that had gone wrong in his life starting with his relationship with Katherine. This conversation happened in the first episode of the show and pretty much summed up what the Salvatore brothers' relationship would look like for the rest of the show.
Elena was at the center of almost everything that went wrong between the brothers from the moment Damon showed up in Mystic Falls. While Elena wasn't always the problem, both brothers loved her, and whenever she was around, both of them just couldn't behave normally. Their time in the show was largely shaped by decisions they made based on their feelings for her.
"Whenever You Go Too Far, I Will Be There To Pull You Back."
"1912" (Season 3, Episode 16)
Damon was reassuring Stefan of his after the wrangles they had been through in the earlier seasons. In the first season, Damon was the villain and Stefan was the one doing the pulling back, but that had changed by the third season. Damon regretted having let Stefan fall over the rails in 1912 and believed he could have prevented him from turning into a monster.
This was one of the early statements that redeemed Damon, proving that he was capable of being a caring elder brother. Stefan was also a good brother whenever Damon needed him. Despite their frequent squabbles, the brothers were always there for each other whenever one of them went too far.
"You're Choosing That Woman Over Me?"
"The Simple Intimacy Of The Near Touch" (Season 8, Episode 9)
The worst times for the brothers came whenever they let evil people get into their heads and divide them. In this case, Sybil had caused a rift between the Salvatore brothers and Damon was only reminding Stefan that it wasn't the first time he chose a woman over his own brother. Both Damon and Stefan had poor character judgment (especially when they thought they were in love), which is why Sybil and Katherine managed to divide them.
In Katherine's case, Damon was supposed to be the wise elder brother, but he let his feelings cloud his judgment. Sybil was way more dangerous than Katherine though, and both brothers failed to work together against her from the start leading to another dangerous rift that nearly ended the both of them.
"We Team Up, ... We Activate Our Wonder Twin Powers."
"1912" (Season 3, Episode 16)
Damon was reminding Stefan of the great team they made back in 1912 after they were turned into Vampires. Stefan, on the other hand, was too afraid of going back to drinking human blood because he felt guilty for having lost it and snapped a woman's neck.
Damon was still right about making a good team because the brothers were at their strongest only when they worked together. While they did lots of good things when they worked together, the teaming up that Damon was suggesting would always lead Stefan down the wrong path, which is why it wasn't the best advice.
"This Is Not How You Are Going To Die, Stefan Salvatore."
"I Went To The Woods" (Season 7, Episode 17)
The Phoenix Stone was one of the greatest threats the Salvatore brothers faced and when they got trapped inside, they had to face their greatest fears. This conversation was Stefan's key to getting out of the stone and while it was in his mind, it expressed the depth of Damon's love for him. Despite their differences, the brothers were also each other's best allies against deadly threats.
Damon wasn't always a good brother to Stefan, but when they were faced with real peril, he was always there to help. Stefan was mostly the one giving the encouragement because Damon went off the rails more frequently, especially when he turned off his humanity.
"... When I’m Not A Good Brother People Get Hurt."
"I Would For You" (Season 7, Episode 15)
"I'm not a good brother when I'm not with Elena..." Damon's quote starts. Damon was considering making himself the new subject of Rayna's wrath in Stefan's place. His statement was true because Damon wasn't exactly a model elder brother.
When the brothers hunted together in the flashbacks, Damon was the most ferocious of the two and killed even more people when he turned off his humanity. He was also at fault for most of the deaths in Mystic Falls that happened at the beginning of the show. Stefan was often the good brother trying to save the world from Damon.
"...I'm Here To Forgive You, Damon"
"Nostalgia's A B**ch" (Season 8, Episode 10)
Damon hurt Stefan more times than Stefan did to him, but Damon was never good at saying sorry which is why Stefan had to learn to forgive without Damon apologizing.
This was the conversation between the brothers after Stefan entered Damon's head in a bid to save him from Sybil's control. Everyone had tried to save Damon but failed because Stefan was the only person that could get into his subconscious. The conversation summarized the tumultuous lifetimes the brothers had spent together since they became vampires.
"... You Have To Want To Be Human For Yourself."
"I'll Wed You In The Golden Summertime" (Season 6, Episode 21)
Damon was about to take the cure that would turn him into a human forever and wanted his brother's insight on whether it was the right thing to do. It was just one of the many times when Damon needed a pat on the back and some reassurance and Stefan was always the wise person to give it.
Stefan knew that it was a difficult decision for Damon, and he advised him that he should make the decision himself and not rely on what Stefan and Elena thought about it. The brothers were always there for each other whenever they needed to make life-changing decisions.
"You Will Be With Me... To Destroy Everything... I Build."
"Black Hole Sun" (Season 6, Episode 4)
Stefan was reminding Damon that his very presence was enough to ruin anything good that the younger brother tried to build. When Stefan came back to Mystic Falls, everything was alright and Stefan was happy with Elena until Damon showed up and destroyed everything.
Stefan was always the cool-headed brother that tried to do the right thing, including cleaning up after his elder brother. If Damon had stayed out of Stefan's way, Katherine would not have caused the brothers so much pain. Damon made the same mistake with The Originals and Sybil and Stefan got caught up in the mess.
"You're Here Because You Blame Yourself For Turning Me."
"Nostalgia's A B**ch" (Season 8, Episode 10)
Stefan had come into Damon's mind to save him from Sybil, but Damon wasn't willing to accept that the troubles the brothers had been through were partly his fault. While Stefan made mistakes in the past, he had spent his whole life cleaning up after Damon.
Damon never stopped blaming Stefan for his becoming a vampire which was unfair because taking Katherine's blood willingly was one of the worst decisions Damon ever made. Despite Stefan's frequent efforts to repair their relationship, Damon would always bring up the past to make Stefan feel guilty.