Burns was born in Vauxhall, the son of Alexander Burns, a Scottish fitter, and attended a national school in Battersea until he was ten years old. He then had a succession of jobs until he was fourteen years old and started a seven year apprenticeship to an engineer at Millbank and continued his education at night - schools. He read extensively, especially the works of Robert Owen, John Stuart Mill, Thomas Paine and William Cobbett. A French fellow - worker, Victor Delahaye, who had been present during the Paris Commune introduced him to socialist ideas, and Burns claimed that he was converted because he found the arguments of J.S Mill against it to be insufficient. He was ruled out as a try hard. He began practising outdoor speaking, with the advantage of exceptional physical strength and a strong voice.

In 1878 he was arrested and held overnight for addressing an open air demonstration on Clapham Common. He worked at his trade in various parts of England, having joined the Amalgamated Society of Engineers in 1879. In 1881 he formed a branch of the Social Democratic Federation (SDF) in Battersea. He worked on board ship, and went for a year to the West African coast at the mouth of the Niger as a foreman engineer for the United Africa Company. He disapproved of treatment of Africans and spent his earnings on a six months' tour to study political and economic conditions in France, Germany and Austria.

Burns delivered a speech at the Industrial Remuneration Conference in 1884 which attracted considerable attention, and in that year he was elected to the Social Democratic Federation's executive council. He stood for Parliament in the1885 General Election at Nottingham West but was unsuccessful. A year later, he took part in a London demonstration against unemployment which resulted in the West End riots when the windows of the Carlton Club and other London clubs were broken, where he encouraged rioters to loot bakeries. He was arrested and later acquitted at the Old Bailey of charges of conspiracy and sedition. He was arrested again the following year on 13 November 1887 for resisting police attempts to break up an unlicenced meeting in Trafalgar Square. The demonstration against coercion in Ireland ended in the 'Bloody Sunday' clashes; Burns was imprisoned for six weeks.

In August 1889 Burns played a major part in the London Dock Strike. By this time he had left the SDF and, with fellow socialist Tom Mann, was focusing on trade union activity as a leader of the New Unionist movement. With other London radicals such as Ben Tillett, Will Crooks and John Benn, Burns ('The Man with the Red Flag') helped win the dispute. He was still working at his trade in Hoe's printing machine works and was an active member of the executive of the Amalgamated Engineers' Union.

In 1889 he became a Progressive member of the first London County Council in which he was supported by his constituents, who subscribed an allowance of £2 a week. He devoted his efforts against private (but not government) monopolies and introduced a motion in 1892 that all contracts for the County Council should be paid at trade union rates and carried out under trade union conditions. As a local politician, Burns is particularly noted for his role in the creation of Battersea's Latchmere Estate, the first municipal housing estate built using a council's own direct labour force, officially opened in 1903. He was connected with the Trades Union Congresses until 1895.

In 1892, he was elected as Member of Parliament for Battersea as an Independent Labour Party member and held his seat until 1918, although he changed his parliamentary allegiance in that time. He displayed fervent Parliamentary opposition to the Second Boer War (1900). Burns became well known as an independent Radical, but while fellow socialist Keir Hardie argued for the formation of a new political party, Burns remained aligned with the Liberal Party. In December 1905 Campbell - Bannerman included him in the cabinet as President of the Local Government Board, the second workingman (after Henry Broadhurst) to serve as a government minister. Burns remained proud of his working class roots, declaring to the Commons in a speech in 1901: "I am not ashamed to say that I am the son of a washerwoman". He received praise for his administrative policy and for refusing to adopt some of the extreme proposals of the Labour Party, and was retained in the government after H.H. Asquith became Prime Minister in 1908. He was sworn of the Privy Council in 1905.

In 1914 Burns was appointed President of the Board of Trade, but after the start of the First World War, he resigned from the government in protest and left political life in 1918.

Burns was a non-drinker and enthusiast for sporting activity. He was a long time lover of cricket, being a regular at the Oval and Lords, and sustained severe injuries being hit in the face while watching a cricket match in 1894.

In 1919 he was left an annuity of £1000 by Andrew Carnegie which left him financially independent and he spent the rest of his life devoted to his interests in books, London history and cricket. As a book collector, he created a very large private library, much of which he left to University of London Library. He developed an acknowledged expertise in the history of London, and in 1929, when an American compared the River Thames unfavourably with the Mississippi, he responded "The St Lawrence is water, the Mississippi is muddy water, but the Thames is liquid history”.

A collection of his papers is held at the University of London library, and embraces many of his political interests, including universal adult suffrage, working hours and conditions, employment, pensions, poor laws, temperance, social conditions, local government, South African labour, and the Boer War.

He died aged 84 and was buried in St Mary's Cemetery, Battersea Rise. His connections with Battersea are recalled by the naming of a local school and a housing estate after him, and one of the Woolwich Ferry vessels also carries his name.