In my understanding, the anti-hero is usually someone who isn't heroic in the standard sense because he lacks some of the striking "good" qualities of a hero, while he has the heroic qualities of strength or purpose or something that gives him power. And he uses that power (and perhaps some underhanded "unheroic aspects") to do something good. That is, his character might not be heroic, and his motivation might not be heroic (he might want the right thing for the wrong reasons, like revenge or money), and his methods might not be heroic, but what he achieves is heroic (the results). And for whatever reason (this is tricky to accomplish), the reader identifies with him in some way.
The classsic pairing has been Superman (hero), Batman-Dark Knight (anti-hero). Superman does the right thing for the right reason. Batman does the right thing for the wrong reason. Both use heroic strengths to achieve something good, but Batman also uses bad tactics.
Scarlet O'Hara is an anti-heroine in that she has heroic strengths that help her survive the war but more than that, to help others (her whole family, her beloved's wife that she hated ever) survive the war. She's willing to do anything to save her farm, including marrying her sister's fiance (not heroic, but achieved a good end). She's actually more of an anti-hero than Rhett (who actually realizes that his romantic need to be a hero in the war was pretty stupid).
There was a time that "antihero" was used to refer to "nebbishy guys" like Benjamin in The Graduate, who have no heroic strengths and aren't at all "Bad", but that didn't last, fortunately. They aren't anti-heroes but more like "everymen".
The Byronic hero is generally considered to be artistic and "moody" and obsessed with women.
I wouldn't say they're anti-heroes, but what we
romance writers would call "the romantic hero"-- heroically tormented
But the antihero usually is the one who does the right thing for the wrong reasons (but with some strength or power that would ordinarily be considered heroic). A villain will often do use heroic strengths to do the WRONG thing (though it might be for the "right" reasons like religion or patriotism).