Damon is one of the most complex characters in The Vampire Diaries. I love many of the other characters but they have less layers, less tragic circumstances. He's also not only a bad boy but essentially the underdog of the series. I have a weakness for both the bad boy with the heart of (tarnished) gold and the underdog.
He's also got intelligence and humour, snark and wit. He's loyal when it counts and he'll kill - but also and most importantly, die, for Elena. He loves so deeply that it hurts even when it is returned. He doesn't show it but he wants to be loved, he wants to be accepted but he's broken thanks to his father, thanks to Katherine and, most importantly I think, thanks to his brother.
He's the man who wants nothing but to give everything of himself but has learned not to because people show him it's not wanted or take advantage. He's a sensitive soul with a hardened shell (he'll always be that way, there's no erasing what he's been through for nigh a century and a half) and if given half the chance, he cold have been the sweetest man alive. He's devoted and impulsive and rash, but he also likes to be the bad boy. He's contradictory.
Like I said, he's the most complicated character on TVD - and that's why I love him.
I could write something here about how his good looks and incredible eyes make him such a good character. However, at the end of the day, wrapping (looks and eyes) do not make a good character--they may make him or her easier to look at it, but that's about it.
I think, the things that separate Damon from most of the cast are the tragic nature of his character, his constant denial of emotions, his love for and loyalty to Stefan and Elena (as two separate entities or together), and, finally, his attempts to conceal the goodness within himself. In my opinion, Damon is the most complex, most tragic character on the show. He has all these layers and this huge facade that he constantly shows.
Also, I think, what makes Damon interesting, is his love and devotion to Elena. He clearly loves her, and even though he may never truly have her (with all this sire bond stuff), he will never stop loving her.
there are so many quirky and yet snarky things about him while staying perfectly cool and collected. The sultry and seductive looks from his eyes and those lips. He is a perfect fit for Damon thru and thru, which is why I in turn fell in love with TVD!
Putting aside his obvious good looks, could the attraction to the Damon character be his internal struggle? The fact that he can't have Elena, and he has to control his vampire impulses, which he does with his humour and salty one liners. These attributes all make this character desirable despite the horrible actions that he does. Damon has a redeeming quality that makes him irresistible, he has the ability to love.
Personally, I love Damon because he is so complex. Although he can be dark, unpredictable and destructive he also has an amazing capacity for love when he allows himself. Think about it he was in love with Katherine for over a hundred years and devoted himself to finding a way to rescue her. He is so passionate and loyal to the people he chooses to love, and yet is never really loved in return. He's a tortured soul and that's why we love him!