Abner is a mentally-ill obese criminal who acted as a henchman of both Two-Face and The Joker, being the only henchman other than Bruno to remain a member of the Joker's squad by the time of the 1970s.
Batman: The Dark Knight Returns
Abner, upon paying the Joker a visit when the latter recovered from his catatonia due to Batman resurging and defeating the Mutant Leader. He then informed the Joker that Two-Face hired him to create explosives to hold the city at ransom, although Joker ordered him to sabotage the explosives so they would prematurely detonate and kill Two-Face. He later appeared when the Joker broke out and killed a TV talk show audience during an appearance that was supposed to show he was cured from his insanity, using human-shaped bombs. He later helped the Joker in selling poisoned cotton candy and warned Joker about the arrival of Batman and Robin. He then, alongside the female bomb, Mary, pursued Robin, eventually strangling her on a roller coaster car, although his recklessness and not minding his surroundings resulted in his downfall, as he hit a low fixture and ended up decapitated.
In other media
He appears in the second part of the two-part animated movie version of the Dark Knight Returns. His role is overall the same, although because of the segment of Two-Face holding the Gotham Life Towers at ransom was moved to before the Joker awakened from his catatonia, his sabotaging Two-Face's bombs to prematurely detonate on the Joker's orders was cut. Similar to the comic, he created Bobby and Mary and also attempted to sell poison cotton candy at a carnival, also notifying Joker about Batman's arrival and pursuing Robin. His death was slightly altered from the comic, however: He attacks her on the tracks, and Robin accidentally kills him by knocking him onto the conveyor spikes on the tracks, and had his head chewed up by the gears onboard.