The second Monday in January is Coming of Age Day, a holiday in Japan to celebrate people who have their 20th birthday and become adults.
"coming of age" is an old English expression meaning:
to become an adult
or
to reach a special age
For example: The Cuban Missile Crisis was JFK's coming of age. = This was the time when people first respected JFK as a real President.
When you become an adult, you can vote ("age of majority") and you can sign legal contracts.
In Canada and the U.S., the age of adulthood is 17 or 18 in most provinces and states.
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Coming of Age Day (成人の日 Seijin no Hi)
This national holiday was established in 1948 as a day to congratulate and encourage people who have reached the age of majority (20) during the year. Cities and towns throughout the nation hold ceremonies for these people.
Originally held on January 15, in 2000 it was changed to the second Monday of January in accordance with the Happy Monday System.