Prior to the Meiji Period, the date of the Japanese New Year was based on the Chinese lunar calendar, just as the contemporary Chinese, Korean, and Vietnamese New Year celebrations are. However, in 1873, five years following the Meiji Restoration, Japan adopted the Gregorian calendar and the first day of January became and continues to be the official and cultural New Year's Day in modern Japan.
Much like Christmas for Christians, Gantan Sai has become a national holiday in Japan and expanded out past the Shinto religious practices. Today is is mostly referred to simply as Japanese New Year or Shogatu. New Year's is celebrated for seven days, though shops are only closed for the first three.
Traditionally, the Shintos visit the shrines, mostly at midnight and pray for the renewal of their heart, prosperity and health in the year to come. It is also common to visit close friends and family to express good wishes. During this time people wear their finest clothes as well.
For more contemporary traditions due to being a national holiday read more on Japanese New Year.