Andrew Jackson's most obvious personality trait in my mind was his ferocious dedication to defending what he saw as his honor. Jackson is famous for his involvement in a number of duels and is also famous for the degree to which he became enraged at the treatment of his wife.
Jackson fought many duels and did so rather violently. For example, there was a duel where Jackson was fighting (with pistols) a man who was a much better shot than he. When the signal was given to fire, Jackson did not. He let the man shoot him. Jackson was badly hurt, but took careful aim (he didn't shoot at first because he was not good at shooting accurately and quickly) and killed the man. This was against the code of duelling, but Jackson did not care. This was one of many instances of him being personally involved in violence touching on his honor.
Jackson was also famous for his devotion to his wife and his fierce defense of her. They married when she was still legally married to her first husband (neither of them knew she was still legally married). Because of this, she was heavily criticized, especially during the presidential campaign. She was accused of being an immoral adulteress, basically. Jackson was always very quick to defend her, sometimes violently.
Jackson's most prominent personality trait to me, then, was this inclination to defend his honor, even to the point of extreme violence.