Davis grew up in the rough and tough, then-predominantly Jewish Brownsville section of Brooklyn. His father owned a candy store during the 1920s, Prohibition days. Davis' job, as a young boy, was to keep lookout for the police and give the alert to his father to hide bottles of whiskey being sold on the sly.
Davis developed into a tough, street-smart young man, and became well known in a neighborhood that was famed as the home of Murder, Inc. His two brothers were affiliated with the notorious gang, acting as collectors. However, because of Al's toughness and fierce independence, he was able to steer clear of the temptations of the easy wealth of the gangster life by concentrating on his boxing. In fact, he was one of the few young men in the neighborhood who was unafraid to stand up to feared local hoodlums like Abe Reles.
His mother called him "Vroomeleh," an affectionate diminutive of his Hebrew name, Avrum (Hebrew for his middle name, Abraham), and he was known to friends and family in his neighborhood as "Vroomy." When Al was a teenager, a boxing promoter convinced him to change his nickname to "Bummy;" the promoter felt that it sounded tougher and would draw a larger crowd. Davis originally objected to his boxing name.
Davis was a rough slugger with a powerful left hook. His record was 66 wins, with 47 KOs, 10 losses and 4 draws, and he was named to Ring Magazine's list of 100 greatest punchers of all time.
Davis made a name for himself when he scored a three-round TKO over the great, but aging former lightweight champion Tony Canzoneri on November 1, 1939. He followed up that victory with a five-round KO over Tippy Larkin on December 15, 1939.
The next year proved a poor one for Davis. First, he lost a unanimous decision to lightweight king Lou Ambers in a non-title match on February 23, 1940. He then fought a memorable non-title fight against welterweight champion Fritzie Zivic on November 15. Zivic knocked Davis down in the first round and thumbed Davis in the eye in the second. Davis then went berserk and hit Zivic with no fewer than 10 foul blows, causing the referee to disqualify him. He then kicked the referee and had to be restrained from attacking Zivic. For this behavior, the New York Boxing Commission disqualified him from boxing in New York State for life, although he was later re-instated. Zivic and Davis then fought a more restrained rematch on July 7, 1941, which Zivic won by a 10th-round TKO.
Davis' last victory over a name fighter came on February 18, 1944, at the expense of former and future NYSAC lightweight champion and future boxing hall-of-famer Bob Montgomery, knocking Montgomery out in the first round. However, he lost a decision to former champion Beau Jack on March 17 of that same year. He also lost by TKO in the second round to former triple titleholder Henry Armstrong on June 15, 1944.
Davis' last big fight came against future middleweight champion Rocky Graziano, who TKOed him in the fourth round of a May 12, 1945 match.
On November 21, 1945, Davis was drinking beer at Dudy's Bar in Brownsville. He had recently sold the bar, and was drinking with his friends when four armed robbers walked in. Davis attacked the robbers, knocked one of them down, was shot three times, but still managed to chase the other three. During the pursuit, he was shot a fatal fourth time.
On November 22, 1945, a headlined article about his death, including a photograph of Davis, was on the front page of The New York Times.