Short answers, pulled from the story.
Annates were payments made by the recipient of a church office to the authority that appointed them, ultimately amounting to half or all of the first year's profits of that office. They were also called "first fruits" (primitiae), a term shared with ancient Greek, Roman, and Hebrew religious offerings. After the papacy claimed the right of appointment for many benefices, the payments flowed into the papal treasury, the Apostolic Camera.
A custom of paying fees to the ordaining bishop dates from the sixth century. The formal right of a bishop to claim the first year's profits from a newly inducted priest is first recorded under Pope Honorius III, who died in 1227. Pope Clement V extended the claim to all vacant benefices in England in 1305, and Pope John XXII extended it across all of Christendom in 1319.
The four classes were the servitia communia (one year's revenue owed by abbots, bishops, and archbishops on induction), the annalia or jus deportuum (due on benefices reserved by the church for the papacy), the quindennia (paid every fifteen years by corporate bodies under a 1469 bull of Paul II), and the servitia minuta (a small notarial fee added to the others).
In his 1520 work To the Christian Nobility of the German Nation, Luther wrote that the pope had "broken the compact, and made the annates a robbery, to the injury and shame of the whole German nation." He called on every prince, nobleman, and city to forbid their subjects from paying annates to Rome.
The Act in Restraint of Annates was obtained by Thomas Cromwell from parliament in 1534. It redirected annates away from the papal treasury and made them payable to the English Crown instead. A new valuation was established by the commissioners who compiled the King's Books (Liber Regis) in 1535.
In February 1704, Queen Anne granted the annates collected by the Crown to support poorer clergy in England, a scheme established by the Queen Anne's Bounty Act 1703 (2 and 3 Ann. c 20). Because the act continued to use the 1535 Tudor-era valuations, the payments remained frozen. By 1837, the Ecclesiastical Commissioners reported first fruits bringing in only £4,000-5,000 a year, against a true value estimated at over £150,000.