Short answers, pulled from the story.
Gregorio Allegri (c. the 14th of January 1582 - the 17th of February 1652) was an Italian Catholic priest and composer of the Roman School. He is chiefly known for his Miserere mei, Deus, a setting of Vulgate Psalm 50 for two choirs that was performed annually during Holy Week in the Sistine Chapel.
The Vatican wanted to preserve the work's aura of mystery and inaccessibility. They threatened anyone who published or attempted to copy it with excommunication. The restriction held for over a century until Mozart transcribed it from memory in 1770.
In 1770, a fourteen-year-old Mozart traveled to Rome with his father and heard the Miserere twice in the Sistine Chapel. He then wrote it out entirely from memory, creating what is considered the first known unauthorised copy. There is also evidence that copies had already been circulating in Europe before his visit.
Dr Charles Burney, an English traveller and music historian, obtained Mozart's transcription and published it in England in 1771. However, his edition did not include the ornamentation for which the work was famous.
Allegri's music for the Sistine Chapel descended from the Palestrina style, and in some cases stripped that refined style of almost all localised ornamentation. While many categorise the Miserere as stile antico or prima pratica, its polychoral techniques make Giovanni Gabrieli a more accurate comparison.
A copyist in the 1880s introduced an error that persists in performances today. The shift from G minor to C minor that audiences now hear reflects the second half of each verse transposed up a fourth. The original version of the Miserere never included the famous top C.