Marion Greenwood was born on 24 August 1862 in Hull, the daughter of George Greenwood, a shipping agent, and Agnes Hamilton. The family moved to Oxenhope, Yorkshire, when she was seven. She was educated at Bradford Girls' Grammar School and came to Girton in 1879 to study natural sciences. One of the first women to undertake independent research in Cambridge, she began research in physiology, specifically the digestive processes, in 1883, with a Bathurst research studentship from Newnham College. She was also the first holder of Girton’s Gamble Prize in 1888.

In 1884 Marion Greenwood was appointed a demonstrator and lecturer in physiology and botany at the then new Balfour Laboratory for women students in Downing Place. She was head of the Balfour Laboratory from 1890 to 1899. She also undertook demonstrations in the university laboratory and supervised Newnham and Girton students in physiology, regularly walking between the two colleges. She continued her research in spite of her teaching and administrative commitments. In 1895 she became the first woman to give a paper in person to the Royal Society (on food vacuoles in infusoria, published in Philosophical Transactions of the Royal Society).

Marion Greenwood had wide literary and domestic interests in addition to scientific ones. She enjoyed cooking, and examined in cookery for the Gloucestershire School of Cookery and Domestic Economy. In addition to her scientific publications, she co-authored, with Florence Baddeley, 'Domestic Economy in Theory and Practice: a Textbook for Teachers and Students in Training' (1901).

Marion Greenwood gave up teaching and research in 1899 soon after she married George Parker Bidder (1863-1953), a marine biologist. They lived in Plymouth for a time, returning to Cambridge in 1902. Their daughters, Caroline Greenwood Bidder and Anna McClean Bidder, born in 1900 and 1903, studied at Girton and Newnham respectively. Anna, who became a marine zoologist, was a founder and first president of Lucy Cavendish College, Cambridge.

Marion Bidder had a wide range of interests and public duties and the family home at Cavendish Corner on Hills Road in Cambridge was a centre of hospitality to many people. She was president of the Cambridge Women's Liberal Association and a keen supporter of women’s suffrage. While at Newnham she had organised a Sunday class for working men in Barnwell. She was an associate of Newnham College, 1893-1904, and member of council, 1894-96. For many years she was a Poor Law Guardian and she was vice-chairman of the Cambridgeshire Voluntary Association for Mental Welfare, the first of its kind in the country. She sat on the Women's War Agricultural Committee, 1916-17; she was a governor of Homerton Teacher Training College for eighteen years and chairman of its trustees from 1925; and a governor of Girton College, 1924-32.

Marion Greenwood Bidder died in Cambridge on 25 September 1932. The centrepiece of Woodlands Court at Girton was given by friends in her memory.