Abbott was born in Asbury Park, New Jersey into a show business family. His parents worked for the Barnum and Bailey Circus: his mother, Rae (née Fisher), was a bareback rider and his father, Harry, was an advance man. Abbott dropped out of school as a child and began working at Coney Island. When Abbott was 16, his father, now an employee of the Columbia Burlesque Wheel, installed him in the box office of the Casino Theater in Brooklyn. Eventually, Abbott began putting together touring burlesque shows. In 1918, he married Betty Smith, a burlesque dancer and comedienne. Shortly after his marriage, Abbott and his new wife began producing a vaudeville "tab show" called Broadway Flashes. This show toured on the Gus Sun Vaudeville Circuit. Around 1924, Abbott started performing as a straight man in an act with Betty. As his stature grew, Abbott began working with veteran comedians like Harry Steppe and Harry Evanson.

Abbott crossed paths with Lou Costello in burlesque in the early 1930s. Abbott was producing and performing in Minsky's Burlesque shows, while Costello was a rising comic. They formally teamed up in 1936 and performed together in burlesque, vaudeville, minstrel shows, and cinemas.

In 1938, they received national exposure for the first time by performing on the Kate Smith Hour radio show, which led to the duo appearing in a Broadway musical, The Streets of Paris. In 1940, Universal signed Abbott and Costello for their first film, One Night in the Tropics. Although Abbott and Costello were only filling supporting roles, they stole the film with their classic routines, including an abbreviated version of "Who's On First?" A common misconception is that Abbott and Costello are the only two non-baseball players who have been inducted into the Baseball Hall of Fame. The comedic duo are not members of the Cooperstown society anymore than the sports writers and broadcasters who are acknowledged by separate awards. However, a plaque honoring and a gold record and transcript of their famous sketch has been included in the museum collection since 1956, and the routine runs on an endless loop on TVs at the Hall, making them one of the few non-baseball players or managers to have a memorial in the Baseball Hall of Fame.

During World War II, Abbott and Costello were among the most popular and highest-paid stars in the world. Between 1940 and 1956 they made 36 films, and earned a percentage of the profits on each. They were popular on radio throughout the 1940s, primarily on their own program which ran from 1942 until 1947 on NBC and from 1947 to 1949 on ABC. In the 1950s they brought their comedy to live television on The Colgate Comedy Hour, and launched their own half-hour series, The Abbott and Costello Show.

Norman and Betty Abbott, the children of Bud Abbott's sister Olive, started on their own careers with help from their uncle: Betty as the script girl on Breakfast at Tiffany's and Norm directed live TV. After Olive's husband abandoned his family, allegedly going for a pack of cigarettes and never coming home, Abbott supported them. He changed everyone's name back to Abbott and raised them as his own children. He also adopted two children with his wife Betty.

Abbott's great-grandniece and granddaughter of Norman Abbott, Kathleen Abbott aka Lisa Bay, was born to Chrissy Abbott in 1966, while Chrissy was attending Beverly Hills High School, and is the adopted sister of director Michael Bay.

Relations between the two partners had been strained for years. In their early burlesque days, their salaries were split 60%-40%, favoring Abbott, because the straight man was always viewed as the more valuable member of the team. That was changed to 50%-50% after they became burlesque stars.

However, other accounts state that the 60%-40% split was Costello's idea. "A Good Straight Man is hard to find" is attributed to Costello. Yet, the sixty-forty split had long irked Costello. Later, after Buck Privates made them movie stars, Costello insisted that the split be reversed in his favor, and it remained sixty-forty for the remainder of their careers. Costello's other demand, that the team be renamed "Costello and Abbott," was rejected by Universal Studios. The result was a "permanent chill" between the two partners, according to Lou's daughter Chris Costello, in her biography Lou's on First. The partners' relationship was also strained by Abbott's battle with alcohol, which began when he took to heavy enough drinking in order to combat the effects of epilepsy. (To make things worse, Costello's wife also had a problem with alcohol.)

The team's popularity waned in the 1950s, and they were further bedeviled by tax issues—the IRS demanded heavy back taxes, forcing the partners (both of whom had been serious gamblers) to sell most if not all of their assets (including Costello's rights to their television show). Abbott and Costello parted ways formally in July 1957. Lou Costello died on March 3, 1959.

Abbott attempted to begin performing again in 1960, with a new partner, Candy Candido, and received good reviews. But Abbott called it quits, remarking that "No one could ever live up to Lou." On TV, he performed in a dramatic episode of General Electric Theater titled "The Joke's On Me" in 1961. A few years later, Abbott provided his own voice for the Hanna-Barbera animated series Abbott and Costello, with Stan Irwin providing the voice of Lou Costello.