Michelangelo became very angry with a certain Cardinal while he was painting the Sistine Chapel. Who was the Cardinal, what was their dispute, and how did Michelangelo express his dislike of the man?

The Last Judgment was an object of a heavy dispute between Cardinal Carafa and Michelangelo: the artist was accused of immorality and intolerable obscenity, having depicted naked figures, with genitals in evidence, so a censorship campaign (known as the "Fig-Leaf Campaign") was organized by Carafa and Monsignor Sernini (Mantua's ambassador) to remove the frescoes. When the Pope's own Master of Ceremonies, Biagio da Cesena, said "it was mostly disgraceful that in so sacred a place there should have been depicted all those nude figures, exposing themselves so shamefully, and that it was no work for a papal chapel but rather for the public baths and taverns," Michelangelo worked da Cesena's semblance into the scene as Minos, judge of the underworld. It is said that when he complained to the Pope, the pontiff responded that his jurisdiction did not extend to hell, so the portrait would have to remain.
https://en.wikipedia.org/wiki/The_Last_Judgement_%28Michelangelo%29

Yes, that taught him to try to censor well known artists.

In fact, the painting WAS "censored" some years later. The Cardinal in question was the ascetic Giovanni Pietro Carafa, later the Italian Grand Inquisitor and Pope Paul IV (1555-59). By the time his papacy ended, the Church was deep into the conservative Counter-Reformation period, and artist Daniele da Volterra was commissioned to paint draperies on the nudes in the "Last Judgment. These could not be removed when attitudes later became more liberal, because Daniele had actually removed the original paint and plaster in parts and refrescoed.

Interesting, these confused censors also had their way with many of the male nude statutes in the Vatican who were given quite painful "cosmetic surgery".