Ever since the Meiji
restoration, the Japanese use mostly the western calendar, i.e. the same system of months and weeks, with the new year beginning at the first of January.
But there's one big difference: They retained their old system of "eras", which correspond to the ruling periods of each emperor. When a new emperor is inaugurated, he declares a new era, and the current year becomes year one of that era, which lasts until his death or abdication. The name of the era is choosen by the emperor and is usually some sort of motto. The current era is Heisei, before that came Showa.