In 1582, Pope Gregory XIII decreed a modification to the Julian calendar. The reform consisted of a one-time correction in the date (skipping some days in the calendar) along with some tweaks to the rules of the calendar itself. For the civil calendar, the only substantive change was to omit 3 leap days every 400 years (in years evenly divisible by 100 but not 400, e.g., 1700, 1800, 1900, but not 2000). There were also changes to the way that the age of the moon was calculated for the purposes of finding the date of Easter. Considered as a theoretical calendar (i.e., projecting it back before it was actually invented), the Gregorian calendar matches up with the Julian in the third century CE.
From a seasonal point of view, a the difference between the Gregorian and the Julian calendar wasn't yet really all that pressing a matter, only 10 days in 1582, only 14 days in the twentieth century. The logical question thus becomes, why did it matter? The short answer is religion, which is, in most cultures, very concerned with keeping time. For Christianity, the fundamental chronological problem was the calculation of Easter, and this difficulty drove the reform. Recall that Easter is supposed to be the first full moon after the spring equinox. But as Easter was calculated by rule, not astronomical observation, the slip of the equinox meant that the rule was badly in error. The Gregorian calendar was designed to restore the spring equinox to the March 21 date that had been traditional since the Council of Nicaea (see Alexandrian Easter).
Most countries adopted the Gregorian calendar in a single, one-time correction. Catholic countries quickly adopted the reform when Pope Gregory proclaimed it. Many of the major countries (Italy, Spain, Portugal, Poland), skipped from Thursday, October 4, 1582 to Friday, October 15, 1582, which we now take, for simplicity's sake, to be the canonical point of switch. Notice that the day-of-the-week cycle is not altered in the switch (October 15, 1582 would have been a Monday in the Julian calendar). The reformers wanted to preserve the seven-day cycle unaltered, as they believed it was biblically ordained. The other Catholic countries quickly followed suit. Some, like France, by the end of 1582, others, like the Catholic parts of Switzerland (and, interestingly, the Spanish colonies in America—probably the result of delays in communication—waited until 1584. For some odd reason, the Spanish Netherlands switched over at the very end of 1582 (from December 21 to January 1), which means they skipped Christmas that year. (The October time was originally picked to omit as few feasts as possible from the church calendar, and Christmas seems a doozy of a feast to skip.)
In an age of intense religious passion, the simple fact that the Pope instituted the reform was enough to make Protestant countries reject the change. The greater part of protestant Germany did not switch to the Gregorian calendar until 1700, the Protestant Cantons of Switzerland and Protestant Netherlands until 1701.
The Swedish dithered. In 1700 they began what was intended to be a gradual switch to the Gregorian calendar. They planned to stop observing leap years until their calendar was in line with the Gregorian one. They did omit the leap year in 1700, but observed the leap year in 1704 and 1708 (apparently they forgot the plan). Thus they were 10 days out of step with the Gregorian calendar and 1 day off from the Julian. Then, in 1712, they changed their minds, and went back to the Julian system by adding two leap days to February. Somewhere in Sweden, there are probably some unique baptismal records of people whose birthday was on a date never to be seen again: February 30. Lithuania and Lativia, which were under Polish rule at the time of the reform (and hence changed in 1582), actually reverted to the Julian calendar, so strong were the feelings. They did not change back again until the 20th century.
Although Queen Elizabeth I initially expressed some interest in changing the calendar in 1582, the Church of England effectively tabled the idea, which was not taken up again for nearly 170 years. By that time, passions had sufficiently cooled that when the idea was introduced as an act of Parliament in 1752, it passed with hardly a murmur. The English colonies in America changed at the same time. By that time, 11 days now had to be added, which the English did by skipping from September 2 to September 14, 1752.
In some areas of Europe, there were riots when authorities tried to introduce the Gregorian calendar (in the 16th century), but the notion that there was popular discontent in England over the shift seems to rest entirely upon a William Hogarth print, which shows a mass demonstration through a window, with the protesters holding up the famous banner saying "give us back our 11 days." Contemporary newspapers and other records, however, give no such indication, although there are some surviving sermons that indicate the authorities took pains to explain the situation carefully so that there should be no misunderstanding. The entire idea probably rests on Hogarth's rather jaundiced view of lower-class ignorance, rather than historical reality.
In 1753, Sweden finally caved in, following Great Britain's lead.
The last Christian countries to accept the Gregorian calendar were the Orthodox ones. Many (those under Russian domination), did not do so until the Bolsheviks decreed the change in 1918 (the October revolution actually took place in November, according to the Gregorian calendar), although there was a history of failed attempts at reform in the 19th century. The Greeks didn't switch over until 1923.