Over the course of The Vampire Diaries, Elena dated both Salvatore brothers, but when did her heart first beat for Damon? They shared their first kiss in the season 2 finale and didn't become more intimate until season 4 after she'd become a vampire. In season 6 after Damon and Bonnie become trapped in a prison world, Elena admits exactly when she first realized she had feelings for the eldest Salvatore.
Elena first dated Stefan and thought Damon was vile. Whereas Stefan lived a life in repentance of what he'd done prior, Damon had no such qualms - until he met Elena. Damon changed the most over 8 seasons of The Vampire Diaries and it's clear Elena was the largest influence on that evolution. Damon professed desire for his future wife, Elena, long before she reciprocated those feelings, but it turns out, not before she first felt them.
Damon admitted his love for Elena in The Vampire Diaries season 2, episode 8, "Rose", when he returned Elena's vervain necklace to her. Before handing the necklace over, Damon compelled Elena to forget his confession. Perhaps he shouldn't have, because Elena started pining for her boyfriend's brother back during season 1 of The Vampire Diaries. According to Elena's own account of things, she knew she had feelings for Damon in season 1, episode 19, "Miss Mystic Falls" when Damon stepped in to act as Elena's escort after Stefan failed to show.
It's obvious, in retrospect, that Damon and Elena's first dance was so significant, and explains why Damon and Bonnie never dated like in the books the series is based on. Damon and Elena couldn't take their eyes off one another. Their chemistry was palpable, and Damon showed Elena that day he would be there for her when Stefan couldn't. Damon ultimately tamed himself so he could be a partner to Elena, but doubt of his intentions lingered among Stefan and their friends for some time.
In fact, that moment in The Vampire Diaries' history was important for another reason. Elena didn't sleep with Damon until after she'd been turned into a vampire, which led to questions of whether she did so of her own volition. Because Damon was Elena's sire, as a vampire she was compelled to do what he wanted and feel what he wanted. Stefan assumed Damon's sire hold over Elena was what prompted Elena to choose Damon, but in reality, she fell for Stefan's brother long before becoming a vampire.
Other important moments between Elena and Damon's romance would come, such as the reveal in the season 3 finale that Elena actually met Damon before Stefan. Even intentionally losing her memories of Damon after he became trapped in the prison world wasn't enough to keep Elena from him. In the end, the love triangle pattern between dueling brothers had to end. First initiated a century ago with Elena's doppelganger, Katherine, who Nina Dobrev also played, it was Damon who won Elena's heart. The Vampire Diaries ended with the promise that Elena and Damon received their happily ever after. Damon took the cure and got to live a human life with Elena, the woman who first fell for him at the Miss Mystic Falls pageant.