The detail
The immediate cause was political and dynastic. Henry VIII, who had earlier been named "Defender of the Faith" by Leo X (1521) for a treatise against Luther, sought an annulment of his marriage to Catherine of Aragon, who had not given him a male heir, arguing that it was invalid because she was his brother's widow. Pope Clement VII, under pressure from Catherine's nephew Emperor Charles V, did not grant it. Parliament cut off appeals to Rome (Act in Restraint of Appeals, 1533), Thomas Cranmer, the new archbishop of Canterbury, declared the marriage void, and the Act of Supremacy (1534) recognised the king as "the only supreme head in earth of the Church of England".
Under Henry the doctrinal change was limited. Thomas More and Bishop John Fisher were executed (1535) for refusing to accept the supremacy; the monasteries were dissolved (1536–1540) and their wealth went to the crown and nobility; an English Bible was placed in churches (Great Bible, 1539); yet the Six Articles (1539) reaffirmed transubstantiation, clerical celibacy and private masses. Under Edward VI (1547–1553) Cranmer led a clearly Protestant reform with the Book of Common Prayer (1549, revised 1552) and the Forty-two Articles (1553). Mary I (1553–1558) restored communion with Rome, and some 280 Protestants were burned, among them Cranmer (1556).
Elizabeth I's settlement (1559) restored royal authority over the church, with the monarch now styled "Supreme Governor", and imposed a revised Prayer Book through the Act of Uniformity. The Thirty-nine Articles (1563, confirmed by Parliament in 1571) are Reformed on justification, Scripture and the sacraments, while the church retained bishops, a set liturgy, vestments and much of the medieval parish structure. Pius V excommunicated Elizabeth in Regnans in Excelsis (1570), after which Catholics were treated as potential traitors and many priests were executed. Tensions with Puritans inside the Church of England led in the following century to Presbyterian and Congregational separation.
The question of continuity remains disputed. Leo XIII declared Anglican ordinations "absolutely null and utterly void" in Apostolicae curae (1896), mainly because of changes to the ordination rite under Edward VI; the archbishops of Canterbury and York replied in Saepius officio (1897). Anglicans have since been in dialogue with Rome through ARCIC (from 1970) and entered full communion with the Old Catholic churches (Bonn Agreement, 1931) and, later, with several Lutheran churches (for example the Porvoo Communion, 1996).
How each side tells it: in the classic Anglican view the Church of England is both catholic and reformed, the same church that existed before the Reformation, freed from foreign jurisdiction and purified in doctrine. In the Catholic view it was a schism caused by royal will, which later adopted Protestant doctrine and broke apostolic succession. Protestant historians long portrayed the Reformation as welcomed by a people tired of a corrupt church (A. G. Dickens), while revisionist historians such as Eamon Duffy and Christopher Haigh have argued that late medieval English Catholicism was vigorous and that the Reformation was largely imposed from above.