Though initially started on antagonistic terms, their relationship develops throughout the series.
He was the narcissistic evil vampire, who came to town to seek revenge on his younger brotherwho Elena "Vampire diaries elena and damon hookup" to develop an interest in. When everyone had written Damon off as a bad guy who couldn't be redeemed, Elena, who was unafraid of him and willing to call him on his crap, saw something in him that hinted he wasn't entirely evil. And so there marked the start of a tentative friendship.
Over the seasons, their bond continues to grow, from becoming allies to friends, to best friends and, ultimately, lovers. He's always saving her from danger and she's saving him from his dark self, they both care deeply for each other, regardless of any obstacles that may come in their way.