Short answers, pulled from the story.
The Gregorian calendar was created to correct a ten-day drift that had accumulated in the Julian calendar since the First Council of Nicaea in AD 325. The Julian calendar overestimated the solar year by about 11 minutes annually, causing the March equinox to fall on the 10th or the 11th of March rather than the ecclesiastically fixed the 21st of March, which in turn distorted the calculation of Easter.
The Gregorian calendar went into effect in October 1582, following the papal bull Inter gravissimas issued by Pope Gregory XIII on the 24th of February 1582. The first countries to adopt it, including Spain, Portugal, France, and Italy, skipped from Thursday the 4th of October 1582 directly to Friday the 15th of October 1582.
The key mathematical proposals came from Aloysius Lilius, a Calabrian doctor, who devised the revised leap-year rule and a new scheme for calculating Easter's lunar cycle. Christopher Clavius, a German mathematician, expanded Lilius's work into an 800-page defense of the reform and advised Pope Gregory XIII to implement the ten-day correction all at once rather than gradually.
Every year divisible by four is a leap year, except for years divisible by 100, which are not leap years unless they are also divisible by 400. For example, 1800 and 1900 were not leap years, but 1600 and 2000 were. This rule produces 97 leap years in every 400-year cycle and gives the calendar a mean year of 365.2425 days.
Saudi Arabia adopted the Gregorian calendar in 2016, making it one of the most recent countries to do so. Among European countries, Greece was the last to adopt it for civil use, doing so in 1923.
The Julian calendar accumulates an error of about one day every 128 years. The Gregorian calendar's error relative to the mean tropical year is roughly one day every 3,030 years by standard estimates, or one day every 7,700 years when measured against the average astronomical vernal equinox interval near the year 2000.