Short answers, pulled from the story.
Pope Gregory XIII issued the papal bull Inter gravissimas on the 24th of February 1582. This decree replaced the Julian calendar to fix a drift that had pushed the vernal equinox away from its traditional date of the 21st of March.
The Gregorian system modifies leap year rules to create an average year of 365.2425 days by excluding years divisible by 100 unless they are also divisible by 400. This rule reduces the number of leap years from 100 to 97 every four centuries and makes 1800 and 1900 common years while keeping 1600 and 2000 as leap years.
Spain and Portugal decreed the change on the 29th of September 1582 under Philip II who ruled much of Roman Catholic Europe. France adopted the system a few months later when the 9th of December was followed by the 20th of December, but Britain and its empire including the eastern United States did not adopt the calendar until 1752.
Historians often write dates as Old Style or New Style to distinguish between calendars during the transition period because England and Wales began the year on the 25th of March Lady Day until 1752 when it shifted to the 1st of January. This dual dating practice helped avoid confusion when the same event had different dates depending on which calendar was used locally.
January derives from Janus the Roman god of gates doorways beginnings and endings while February comes from Februa meaning purification and is cognate with fever and the Etruscan death god Februus. March honors Mars the Roman war god while April likely stems from a verb meaning to open or the goddess Aphrodite, May takes its name from Maia a vegetation goddess whose root means great, June belongs to Juno the goddess of marriage childbirth and rule, July was instituted in 44 BC as part of Julius Caesar's reforms, August follows July and was established by Augustus in 8 BC, and September through December were originally the seventh through tenth months in Romulus ten-month year before January and February were added later.
The Gregorian calendar has an error of about one day every 3,030 years compared to the mean tropical year which improves upon the Julian calendar that drifted one day every 128 years. Sir John Herschel proposed modifying the system in the nineteenth century to include 969 leap days every 4,000 years instead of 970 but his suggestion remains unadopted.