James Franck Bright was born in London, the son of the physician Richard Bright, who described Bright's disease, and Eliza Follett, sister of lawyer-politicians Sir William Webb Follett QC MP and Brent Follett QC MP. He was educated at Rugby School under Thomas Arnold and at University College, Oxford (matriculated 1851 aged 18, graduated B.A. 1855, M.A. 1858, B.D. and D.D. 1884).
From 1856 he was a schoolmaster at Marlborough College, where he was Head of the Modern Department for sixteen years, under the headmastership of George Granville Bradley. Bright wrote the necessary textbooks himself, including "History of England". Bradley had become Master of University College, Oxford in 1870; he recruited Bright to Oxford as a history tutor in 1872, tutoring at Balliol, New and University Colleges. Bright became Fellow and Dean of University College in 1874, and succeeded Bradley as Master of University College from 1881 to 1906.
Bright wrote a history of Victorian England, "The Growth of Democracy", and biographies of the Holy Roman Empress Maria Theresa and Emperor Joseph II. He was a progressive leader at Oxford, helping to improve teaching standards and arguing that theological degrees could be awarded to non-members of the Church of England. In 1882, Bright was one of the first dons at Oxford University to allow women students to attend his lectures, in University College Hall.
In addition to academic activities, Bright was a member of the Oxford City Council, and Treasurer of the Radcliffe Infirmary. He was also shot by a lady in an incident at University College, but survived.
The Shelley Memorial was installed during Bright's mastership, celebrating the life of the poet Percy Bysshe Shelley (1792–1822), an alumnus of University College. At an opening ceremony on 14 June 1893, Lady Jane Shelley, the widow of the poet's son Percy Florence Shelley (1819–1889), presented the Master with a golden key, giving access to the chamber containing the memorial. Bright described Shelley as "the rebel of eighty years ago", "the hero of the present century", and "a prophet who prophesied good things, and not bad".
He died at Ditchingham, Norfolk, on 23 October 1920.
Some of Bright's sermon manuscripts are held in the University College archives.