The relationship between Damon and Elena is complicated and uncertain. Although Damon's initial interest in Elena is fueled by Katherine's secret manipulations (she admits that she used her superior vampire Power to lure both boys directly to Elena as a punishment for what they did to her), and then by his desire to just take her away from Stefan, but it's obvious that their relationship expands after that. It is unclear whether Damon is ultimately in love with Elena or simply has a growing respect for her, however, leading to speculation in both directions by fans. Damon also repeatedly tells Elena how special she is in terms of her life energy, that she burns more brightly than the average person, so there's also that facet of their relationship - Damon wants to harness and exploit that kind of energy and to have it on his side.
In the original four books, Damon makes moves to woo Elena away from his brother in secret. He attempts to seduce Elena in the school gym with his mind manipulation powers, responds to her confrontation in the graveyard/woods by appearing when she calls him, rescues her when she breaks into Caroline's house to try to steal back her diary, saves both her and Stefan from humiliation (and accusations of murder, in Stefan's case) by replacing Elena's stolen diary with Caroline's own when Caroline is about to read a damning excerpt publicly, and provides assorted other aid. Damon gets himself invited to Elena's home by charming her Aunt Judith and pretending to be Damon Smith, a college student, and coming over for dinner. He also blackmails Elena into sharing blood with him by threatening to kill, or at least feed on, her little sister Margaret if she does not.
Damon substantially backs off once Elena announces her engagement to Stefan, however. He still flirts with her, but he never makes the same moves for her again after that. He takes care of her when she's a vampire and helps her get acclimated to her new existence with fangs, but remains fairly controlled the entire time, and he returns her to Stefan when she regains her senses. Elena thinks that she sees Damon crying for her when she dies at the end of the third book. At the end of book four, Damon leaves Elena and Stefan together, saying that he's not like them and rejecting Elena's offer to stay with her and Stefan.
After Elena's death in book 3, Stefan notes that Damon is preying primarily on blonde girls and suggests that this might be because he's mourning Elena. This is unresolved, of course, but is a very real possibility.
In the 'Return' series, their friendship grows and after he is possessed to get rid of Stefan, he realizes how much she cares about his younger brother and they set out together with Matt. However, Matt sees their friendship growing and returns to Fell's Church. Damon and Elena share blood many times throughout the book and she soon realizes that she has feelings for him.