The second Monday of January is a national holiday in Japan called Seijin no hi (成人の日), Coming of Age Day. Seijin no hi is when people who have or will reach the age of maturity, which is 20 years old right now, between April 2 of the previous year and April 1 of this year get to be recognized as adults.
Each municipality holds their own Seijin-shiki or Seijin ceremony (成人式). Although it may look like a religious ceremony from photos of the event since the young people who participate are mostly wear traditional clothing or wafuku (和服), it is actually a government-conducted ceremony, which means it is strictly non-religious. A lot of people do however go to a Seijin-sai or Seijin Festival (成人祭) at a Shinto shrine, after the official ceremony, to wish for luck and to take photos.
What goes on at a Seijin-shiki?
Not that much really. The basic format is for officials from the municipality (usually the city, town, etc.) to give speeches congratulating the young people assembled for reaching adulthood, and exhorting them to great things, whatever - about what goes on at a graduation ceremony. The main thing is to be there; almost every newly-20-year-old person who attends a Seijin-shiki goes there in full formal dress. (More about the costumes worn below.)
One of the main objectives of the Seijin-shiki or Seijin-sai is of course for the parents to take photos of their grown up kids, as well as for the young newly adult people to take photos of each other. It's also common practice to have a class reunion afterwards with your former middle, high or even elementary school classmates.
Seijin no hi is not that traditional per se, but there is a tradition behind it
The widespread wearing of traditional wafuku, as noted above, makes casual observers believe mistakenly that Seijin no hi and its associated ceremonies and such are hundreds of years old and very traditional. It is actually a post-WWII designated day, established by the government in 1948. (The official age of maturity has been 20 years old since 1879.)
However, the notion of having a celebration when one reaches maturity has been around for a very long time. Both boys and girls, just from from the aristocracy in the beginning by later on by the warrior/samurai class, then later than still by the merchant class, went through a coming-of-age type of ceremony called genpuku (元服). The age at which they had genpuku varied by the period, the class, the family, the gender and so forth - anywhere from 12 to 20, or even later. The original kind of genpuku was for boys only, who got to wear a hat called a toribōshi. The genpuku ritual has generally meant you got to wear your hair in 'adult ways, and wear more grown-up clothing.
Some Shinto shrines and even some municipalities still hold a genpuku ceremony, which is considered a form of Seijin-sai or unofficial religious after-event for the Seijin-shiki. This is one held at Nashinoki Jinja in Kyoto.
The Tōshiya archery exhibition at Sanjūsangen-dō Temple
One of the most colorful and famous Seijin no hi events is the archery exhibition at Sanjūsangen-dō Temple in Kyoto. This is not an official Seijin no hi event, but has become associated with it. It started out as an archery competition in the 17th century between young samurai. Nowadays archery, or kyūdō (弓道), one of the martial arts, is a sport. Participants in the Tōshiya exhibition must be at least at shodan level in kyūdō, as well as being 20 years of age.
Although most photos and videos of the exhibition just show the girls clad in their colorful outfits, boys participate too, in about equal numbers. They don't get featured much because their dark outfits are somber, plus I am guessing a lot of the videographers are men.
Here's a video showing that boys do indeed participate.
Arrows are used as a symbolically objects as well as a motif (as a pattern on kimonos for example) on Seijin no hi, expressing a desire for the young person to have a long, straight future before them. Not 'straight' as in sexual preference, but a future without too many bends in the road.
The typical Seijin ceremony costumes
Seijin no hi is one of the major dress-up days in a Japanese person's life, especially for females. And unlike weddings for example, western style clothes have not made much inroads. So this is a day for dressing in the finest wafuku (traditional Japanese clothing) you can afford. As mentioned above, girls usually dress in furisode (振袖), a type of kimono that is meant to be worn by young, unmarried women and girls. Another garment that is worn by a lot of girls on Seijin no hi is a fur stole, which is a custom that seems to have taken hold in the 1960s or so. This is the one piece of western style garment worn. Nowadays the stole is usually made of fake fur, although some real fur ones are still around. (To me they make the girls look like bunnies.)
Boys usually wear a dark reifuku, a traditional costume with a haori or loose jacket worn over a kimono. If they are going full out, they wear a montsuki (family emblem or crest) haori, and hakama, which are kind of like loose trousers There are exceptions, such as these young men dressed in all white. (Yes white is just as 'formal' a color as black.)
Then there's Kitakyūshū!
But there is one city that does things a bit differently, Kitakyūshū in Fukuoka prefecture. It's become the custom there for everyone participating in the Seijin-shiki to dress up in really loud, colorful outfits, with full on 1980s-90s style "Yankee" hairdos and makeup. By "Yankee" (also written as "yanki" or "yankii") I don't mean Americans from the north eastern part of the United States; it's a slang term for a juvenile delinquents and school dropouts and the 'bad dudes' at school, which came into use around the mid to late 1970s or so. You don't see a lot of "yankees" lately, but dressing up as one has stayed popular in Kitakyūshū.
What happens from 2022 onwards?
On April 1, 2022, the official age of maturity in Japan will be lowered to 18 years of age. This mainly affects voting rights, since you can already drive at age 18. The drinking and gambling ages stay at 20. Also, the age of consent at the moment is 16 for females and for 18 for males, but with the new law it will become 18 for all genders.
The problem though is that the typical 18 year old is preoccupied with university entrance exams or getting a job when they graduate in January. Gofuku-ya, sellers of traditional clothing including kimonos, are especially concerned that a lot of youngsters will simply skip the ceremony. The Ministry of Justice doesn't say which age is the right age for Seijin no hi at all, saying 'it's up to each municipality'.
While official announcements about it haven't been made for the most part, what is very likely to happen is that the Seijin no hi age stays at 20 - this is what many gofuku-ya are recommending. (A lot of people, especially girls and their parents, plan a full year or more ahead for the Seijin no hi outfit, which can cost anywhere for 5 figures for a rental to 7-8 figures if you buy it outright.) We should find out what municipalities decide to do later this year or early in 2021.
Can you go and see a Seijin-shiki?
If you are not a family member or an eligible participant yourself of a Seijin-shiki, you are usually not allowed entry into the official ceremony venue. (Besides the age requirement, you also need to prove you are a current resident of the municipality, were born there or your parents or guardians are residents there.) You can observe the colorful goings on at the informal Seijin-festivals at shrines and temples, but do keep in mind that it's really for the newly adult young people and their families, and don't get too pushy or obnoxious in your photo-taking and such. It's a general rule at any time in Japan to ask permission before taking a pic of someone. If they say no, don't push it.
Until 1999, Seijin no hi was on January 15, and people who were eligible to participate were those who had had or would have their 20th birthday by January 16th. But since 2000 the day has been shifted to the second Monday of January to create a 3-day weekend, and the eligibility now follows the 'school year/grade age' format, where if you're born from April 1 to March 31 you're assigned to a particular grade. (Example: if you are born between April 2, 2013 and April 1, 2014, you will be entering first grade in April 2020.)
Also, just in case you don't know, the official school year as well as the fiscal year in Japan is from April 1 to March 31, not January 1-December 31. ↩