He was born at Skipton in Yorkshire, where his father, the Reverend W. Sidgwick (d. 1841), was headmaster of the local grammar school, Ermysted's Grammar School. Henry himself was educated at Rugby (where his cousin, subsequently his brother-in-law, Edward White Benson – later Archbishop of Canterbury – was a master), and at Trinity College, Cambridge. While at Trinity, Sidgwick became a member of the Cambridge Apostles. In 1859 he was senior classic, 33rd wrangler, chancellor's medallist and Craven scholar. In the same year he was elected to a fellowship at Trinity, and soon afterwards became a lecturer in classics there, a post he held for ten years. In 1869, he exchanged his lectureship for one in moral philosophy, a subject to which he had been turning his attention. In the same year, deciding that he could no longer in good conscience declare himself a member of the Church of England, he resigned his fellowship. He retained his lectureship, and in 1881 was elected an honorary fellow. In 1874 he published The Methods of Ethics (6th ed. 1901, containing emendations written just before his death), by common consent a major work, which made his reputation outside the university. John Rawls called it the "first truly academic work in moral theory, modern in both method and spirit." In 1875, he was appointed praelector on moral and political philosophy at Trinity, and in 1883 he was elected Knightbridge Professor of Philosophy. In 1885, the religious test having been removed, his college once more elected him to a fellowship on the foundation.

Besides his lecturing and literary labours, Sidgwick took an active part in the business of the university, and in many forms of social and philanthropic work. He was a member of the General Board of Studies from its foundation in 1882 till 1899; he was also a member of the Council of the Senate of the Indian Civil Service Board and the Local Examinations and Lectures Syndicate, and chairman of the Special Board for Moral Science.

He was one of the founders and first president of the Society for Psychical Research, and was a member of the Metaphysical Society. Prominently, he took in promoting the higher education of women. He helped to start the higher local examinations for women, and the lectures held at Cambridge in preparation for these. It was at his suggestion and with his help that Anne Clough opened a house of residence for students, which developed into Newnham College, Cambridge. When, in 1880, the North Hall was added, Sidgwick, who in 1876 had married Eleanor Mildred Balfour (sister of A. J. Balfour), lived there for two years. After Clough's death in 1892 Mrs Sidgwick became principal of the college, and she and her husband lived there for the rest of his life. During this whole period Sidgwick took the deepest interest in the welfare of the college. In politics he was a Liberal, and became a Liberal Unionist (a party that later effectively merged with the Conservative party) in 1886. Early in 1900 he was forced by ill-health to resign his professorship, and died a few months later.

Sidgwick was a famous teacher. He treated his pupils as fellow students. He was deeply interested in psychical phenomena, but his energies were primarily devoted to the study of religion and philosophy. Brought up in the Church of England, he drifted away from orthodox Christianity, and as early as 1862 he described himself as a theist. For the rest of his life, though he regarded Christianity as "indispensable and irreplaceable – looking at it from a sociological point of view," he found himself unable to return to it as a religion.

In political economy he was a Utilitarian on the lines of John Stuart Mill and Jeremy Bentham. His work was characterized by its careful investigation of first principles, as in his distinction of positive and normative reasoning, and by critical analysis, not always constructive. His influence was such that for example Alfred Marshall, founder of the Cambridge School of economics, would describe him as his "spiritual mother and father." In philosophy he devoted himself to ethics, and especially to the examination of the ultimate intuitive principles of conduct and the problem of free will. He adopted a position which may be described as ethical hedonism, according to which the criterion of goodness in any given action is that it produces the greatest possible amount of pleasure. This hedonism, however, is not confined to the self (egoistic), but involves a due regard to the pleasure of others, and is, therefore, distinguished further as universalistic. Lastly, Sidgwick returns to the principle that no man should act so as to destroy his own happiness.