Like many countries, Japan uses a 12-month calendar. The names used are very simple. January is literally "Month one" 一月, February is "Month two" 二月, and so on. However, before the Meiji Restoration (mid-1800s), an older 12-month system was common. These months’ names referenced the weather and the seasons, similar to how we name the Full Moons.
December is 師走. The kanji (I think that is the correct term for an ideogram) 師 can refer to a teacher, or a mentor, often in a religious sense. In this context, it means a monk. The second ideogram is 走 which means "running." So, this December literally means "monks running."
But why?
In December, monks are very busy preparing for the New Year's festival, so this last month of the year is "the month of running monks."