In America, most children associate Christmas with some fat dude who climbs into your house and eats your milk and cookies, and in return gives kids gifts which they forget after a few days.
In Italy, children are "visited" by a good witch called La Befana that brings presents and candy to the good children, and coal to the naughty children on the night of January the fifth.
In Iceland, it is believed that during the thirteen days before Christmas, thirteen Yule Lads (gnome-like creatures) enter children's rooms and leave them presents according to each of the dwarves personalities. The most loved Gnome is one that steals your candles, but leaves the best presents.
In Switzerland, two Santas roam the streets during December 6th. One of them is dressed in red and is called Samichlaus, while the other one dresses in black and is called Schmutzli. Samichlaus gives away candy to good kids, while Schmutzli gives sticks to naughty kids.