Refers specifically to January 1st, the first day of the new year in the modern calendar.
元日には初詣に行く人が多い。
Many people go for hatsumōde (first shrine visit) on New Year's Day.
元日は家族で過ごすのが恒例だ。
It's customary to spend New Year's Day with family.
This year, I saw in New Year's Day at a hot spring.
The whole family celebrated New Year's Day together.