The diminutive actor (5 feet 2 inches (1.57 m)) originally wanted to become a Teacher, but failed to gain such employment, and worked in a solicitor's office. He found this job too mundane and he began to take an interest in the theatre. After indulging in amateur theatricals, Brough attended the Royal Academy of Dramatic Art in the mid-1920s. After graduating, he joined a Shakespearean theatrical troupe, where he met his wife-to-be, Actress Elizabeth Addyman. After they married, they used their wedding dowry as collateral to rent the Leas Pavilion, a repertory theatre in Folkestone, Kent. They had one daughter, Joanna, who was educated at Ashford School for Girls.

Following demobilisation, Brough resumed his acting career and reopened the Folkestone rep. Many prominent actors began their careers with the Arthur Brough Players, including Peter Barkworth, who appeared in The Guinea Pig in 1948: Eric Lander, later a star of the TV series No Hiding Place, in 1949: Polly James in the 1960s: and Anne Stallybrass, who started out as ASM in 1960 and went on to play Ida the maid in Pool's Paradise by Philip King; as well as appearing in The Aspern Papers, Candida, and A Taste of Honey at the little Folkestone theatre. Others included Andrew Jack; Sydney Sturgess, who went on to marry Barry Morse; and Trevor Bannister, who would later act alongside Brough in Are You Being Served?

One of the first jobs Brough did away from the stage was the film The Green Man with Alastair Sim, in which he played the landlord of the eponymous hotel. He had a minor role opposite Jayne Mansfield in The Challenge (1960), and made guest appearances in TV shows such as Upstairs, Downstairs (Episode 3.2), Dad's Army, Z-Cars, The Persuaders, Adam Adam Ant Lives!, Randall and Hopkirk (Deceased) and Jason King. He also continued to appear in theatrical productions, including Half a Sixpence (1967), playing a shopkeeper. The Folkestone Rep continued until 1969 before closing at the time that Brough's wife Elizabeth began to suffer ill-health.

In 1972, Brough was cast as Ernest Grainger in the BBC sitcom Are You Being Served? by Jeremy Lloyd and David Croft. Initially a pilot episode in the Comedy Playhouse slot, it was well received and commissioned for a series in early 1973. Set in a fading department store, Brough played the senior menswear salesman, with assistants Mr. Humphries (John Inman) and Mr. Lucas (Trevor Bannister). The show became enormously popular, with an audience of 22 million in 1979, and ran until 1985.

After the show completed its fifth season in 1977, all was going well when, on Easter Sunday 26 March 1978, Arthur Brough's wife of 50 years, Elizabeth, died, and the emotionally devastated Brough announced he was quitting acting. Brough stayed with his daughter for a few weeks following his wife's death and, according to his daughter, Jeremy Lloyd and David Croft made contact to say they were writing him into the next series. However, he died just two months after his wife, on 28 May 1978, in Folkestone. Croft decided not to have another actor take over the part of Mr. Grainger, so his character in Are You Being Served? was replaced by Mr. Tebbs, played by James Hayter.

Brough's daughter, Joanna Hutton, (died 2002) became the first female curator of the Brontë Parsonage Museum in Haworth, for a period in the 1960s.

With a mischievous sense of humour, he would often pull pranks on the rest of the cast during recordings. Despite this, however, Trevor Bannister held him in very high regard, saying of him that he was a "wicked old man but a wonderful man." David Croft recalls the time Arthur would disappear from the set. 'Whenever we were rehearsing he'd vanish at about three minutes to eleven. For a while we wondered where he went, but eventually discovered that he'd nip next door to the pub for a quick Pink Gin. We'd watch from the window as this little figure hurled towards the pub – we never spoke to him about it. One day when he returned, John Inman asked where he'd been. He made some excuse, but what he'd forgotten was that it was pouring with rain and his bald head was soaking wet!'