It's complicated, but a Byronic hero tends to be rebellious, a loner, darkly romantic, and often an antihero.
Shmoop's literature glossary has a short description:
Cooked up by the "mad, bad, and dangerous to know" Lord Byron, a Byronic hero is an antihero of the highest order. He (or she) is typically rebellious, arrogant, anti-social or in exile, and darkly, enticingly romantic. Did we mention Byronic heroes tend to also be kind of hot? Yeah, that too.
TV Tropes has a much longer description, with a long list of defining characteristics:
The Byronic Hero is a type of character popularized by the works of Lord Byron, whose protagonists often embodied this archetype, though they existed before him, it became prominent during Romanticism. Sometimes an Anti-Hero, others an Anti-Villain, or even Just a Villain, Byronic heroes are charismatic characters with strong passions and ideals, but who are nonetheless deeply flawed individuals who may act in ways which are socially reprehensible because he's definitely contrary to his mainstream society. A byronic hero is on his own side and has his own set of beliefs which he will not bow nor change for anyone. A Byronic hero is a character whose internal conflicts are heavily romanticized and who himself ponders and wrestles with his struggles and beliefs. Some are portrayed with a suggestion of dark crimes or tragedies in their past.
The following traits are very characteristic of Byronic heroes and may be helpful in identifying them:
- Is usually male (though female examples are not unheard of) and is always considered very attractive physically and in terms of personality, possessing a great deal of magnetism and charisma, using these abilities to achieve social and romantic dominance. One mark against him personality wise, however, is a struggle with his own personal integrity.
- Is very intelligent, perceptive, sophisticated, educated, cunning and adaptable, but also self-centered.
- Is emotionally sensitive, which may translate into being emotionally conflicted, bipolar, or moody.
- Is intensely self-critical and introspective and may be described as dark and brooding. He dwells on the pains or perceived injustices of his life, often to the point of over-indulgence. May muse philosophically on the circumstances that brought him to this point, including personal failings.
- Is cynical, world-weary, and jaded, often due to a mysterious Dark and Troubled Past, which, if uncovered, may reveal a significant loss, or a crime or mistake committed which still haunts him, or, conversely, that he may be suffering from some unnamed crime against him.
- He is extremely passionate, with strong personal beliefs which are usually in conflict with the values of the status quo. He sees his own values and passions as above or better than those of others, manifesting as arrogance or a martyr-like attitude. Sometimes, however, he just sees himself as one who must take the long, hard road to do what must be done.
- His intense drive and determination to live out his philosophy without regard to others' philosophies produces conflict, and may result in a tragic end, should he fail, or revolution, should he succeed. Because of this, he is very rebellious, having a distaste for social institutions and norms and is disrespectful of rank and privilege, though he often has said rank and privilege himself. This rebellion often leads to social isolation, rejection, or exile, or to being treated as an outlaw, but he will not compromise, being unavoidably self-destructive.
An excellent defining example, although not from literature per se, would be Captain Malcolm Reynolds from the cult TV series Firefly.