Traditionally throughout Europe -- and by transfer of tradition, in the United States as well -- the Winter Solstice has been a time of celebration and gift giving. The reasons are simple; as the shortest day of the year the Winter Solstice symbolizes the point when the harsh world of Winter starts to become the friendly world of Summer again, along with all the promises of better times and new growth that comes with it. Also traditional at these times are the visits of a number of supernatural beings... some friendlier than others. We now largely associate these supernatural visitors with children, but that was not entirely true when the initial stories and traditions began; there is, however, a fairly distinct reason why these supernatural visitors are now expected to be occupied with the children of the household.
In Europe of hundreds of years ago, in order for a family to survive Winter all members of the household had to help. If Winter came and not enough food and resources had been set aside, then a family could literally starve to death before Winter ended! So children who were good and did their chores were a very important thing for helping families to survive the Winter... but, of course, that's not much fun for the kids. I suspect that for this reason, a special set of rewards started to be presented to good children to give them a reason to try their best during the year; and that it should come when most people were already celebrating the change of the weather for the better and exchanging gifts made good sense.
Now, even as a good child could help their family survive Winter, a naughty child who doesn't help could endanger their family... so it also became traditional for warnings to be issued to the troublemakers, often by the same supernatural beings that brought presents though, notably in some cases, said threat of punishment could be delivered by an entirely separate being. And generally a very scary one.
These various beings all appear during different times of the general mid-winter season. The oldest are associated with 'Yule,' a holiday season that once covered from mid-November to January, and which was possibly celebrated thousands of years back... so real ancient monster stuff happening with that! The Yule season (aka "Yuletide") was eventually trimmed down to a shorter three-day holiday that roughly corresponded with the Christian holiday of 'Christmas,' celebrated on December 25. In addition, a number of beings appear during the 'Twelve Days of Christmas' (aka "Twelfthtide" or "Twelfths"), which are the days starting with the holiday of Christmas and ending twelve nights later on January 5.
SO! Let's have a look at some of these supernatural beings, shall we?