Shortest day of the year in Northern Hemisphere is considered 21 December. Climatically it might not be as correct as it is considered.
The history of shortest day of the year goes back to Julius Caesar. Julius Caesar declared 25 December as the date of winter solstice in his Julian calendar. When Christianity gained popularity in Rome, the scholars from Christian world coincides, the birth date of Jesus Christ with 25 December as the day was already observed for celebrations and holidays associated with the winter solstice.
Pope Gregory XIII in Gregorian calendar reassigned the shortest day of the year to 21 December and he kept the Jesus Christ’s birthday at 25.
However the day is longest in terms of daylight in the Southern Hemisphere.
The shortest day of the year occurred when earth tilts farthest at its axial from the sun being at maximum distance of 23 degrees and 26 minutes. The solstice results in the gradual lengthening of nights and shortening of the days in Northern hemisphere. The winter solstice is also the shortest day of the year for the people living at 23-26 N and 23-26 S, at the Tropic of Cancer and Tropic of Capricorn.
The same is true for the southern hemisphere but on a different date and for southern hemisphere it is June 21 and 22.
The shortest day of the year is interpreted differently in both the hemispheres but with one single commonality of recognizing the birth of Christ, festivals and other winter holidays.