After having read too many sources including Wikipedia, I think I can gather what caused the 10 days difference between the Julian and Gregorian calendar, but I'm still unsure of a few details that might lead to a useful discussion. So feel free to add what I might have neglected or needs to be corrected. Also if you have a special calendar in your country that you would like to make an introduction about it, go right ahead.

The Julian calendar introduced an error of 1 day every 128 years. So every 128 years the tropical year shifts one day backwards with respect to the calendar.
Also, the reckoned Moon that was used to compute Easter was fixed to the Julian year by a 19 year cycle. However, that is an approximation that built up an error of 1 day every 310 years. So by the 16th century the lunar calendar was out of phase with the real Moon by four days.
Depended on the fact that it was felt that 21 March was the proper day for vernal equinox (because 21 March was the date for vernal equinox during the Council of Nicaea in AD 325). The Gregorian calendar was therefore calibrated to make that day vernal equinox. By 1582 vernal equinox had moved (1582-325)/128 days = approximately 10 days backwards. So 10 days had to be dropped.

Still there is one thing that I don't understand. Did the accuracy of Julian calendar change after the introduction of Gregorian calendar? It seems so, because dates haven't been changed since that time, Christmas is still being celebrated on Jan. 6th of Gregorian calendar by Eastern Orthodox churches, unless there is still something that I don't know. (JM, could you verify please?)

Speaking of accuracy of calendars, although Gregorian calendar is pretty accurate to maintain the March 21st as vernal equinox, however it is interesting to know that it has larger swings of seasonal errors over the centuries than that of Persian calendar. The New Year in Persian calendar begins at the exact moment of the equinox in the northern hemisphere. The first six months in this calendar are 31 days each, the next five 30 days, and the last month has 29 days but 30 days in the leap years. The reason the first six months have 31 days and the rest 30, was not a random decision by designers– it has to do with the fact that the sun moves slightly more slowly along ecliptic in the northern spring and summer than in the northern autumn and winter. Every 33 years another leap year will be inserted to maintain the seasonal swings, thus the total of 8 leap years in a period of 33 years. Also each month corresponds to only one sign of the zodiac.

Here are the graphs taken from Wikipedia. The horizontal axis represents year in Gregorian Calendar, the vertical axis represents the error in days.

Seasonal error for Gregorian calendar


Seasonal error for Persian calendar