The shortest day of the year is called the winter solstice. The winter solstice occurs on the 21st of December in the Northern Hemisphere.
The length of the day is caused by the tilting of the planet. This makes the sun's path lower in the sky. It also means that the sun does not rise directly in the east nor set exactly in the west. It rises and sets more to the south of those directions. Every day following the winter solstice, the sun moves a little higher in the sky, lengthening the days leading up to the summer solstice in June.