A bad boy allows his interests to take precedence. He can take or leave a relationship with a woman. A bad boy is unattached. Women are not of prime concern to him emotionally.
Some women are drawn to these men who are emotionally unavailable.
A bad boy is a moody charmer. A woman with him won't know what he's thinking or what might trigger good and bad moods. But when he turns his attention to her, she thinks she has won the prize.
His full attention, so hard to get, convinces her she is the one for him.
A bad boy is riddled with paradoxes making him hard to understand. He kicks over a begger's money can, but then he visits his sick aunt in the hospital.
He swerves his bike so he can kill a bunny for dinner, but then he rescues a puppy abandoned alongside the road. He won't remember your birthday, but he volunteers to work with PTSD veterans.
A bad boy has an edge to him. Will he become violent while drinking? Will he pull a knife on someone? Will he move into someone's personal space to menace him.
Will he intimidate someone just to get his way? Will he start brawling or destroy furniture? The protagonist should wonder about his stability.
A bad boy displays an attitude. His attitude is revealed sometimes with clothing, piercings, tats and the like. But even without the accouterments, he oozes Attitude.
That attitude results in treating others poorly. He's so into his own wants and interests that he overlooks those of others.
He's rude, he cheats, he brawls, he lies all because his needs are paramount.
A bad boy is a rebel with or without a cause. He challenges authority just because. He takes short cuts, thinks rules are for other people.
A bad boy likely engages in dangerous hobbies. He might even engage in activities that skirt the law if not outright breaking a law.
A bad boy is mysterious, complex, and complicated. He has a backstory and you may never find out what it is. He carries psychic wounds that are deep and covered up.
He won't bare his soul to a woman or anyone else. Nor does he tend to be self-reflective.
Does every bad boy in books display the panoply of personality described here? No, of course not. But if you create your own mix of traits, you could have a near infinite number of bad boys