Japan is, as you may have guessed, a foreign country. Unless you're Japanese.
One aspect of this is that the many of the holidays celebrated may be completely unfamiliar to non-Japanese anime viewers. Further, holidays known to Western culture may have an entirely different spin on them.
Here is a short chronological listing of Japanese holidays, with an eye to those portrayed in various anime. Note that not all are actual holidays in the sense of "days off". Conversely, unlike in the U.S. with its school-on-Columbus-Day shenanigans, if it's a national holiday in Japan (even if it's just the equinox) you get that day off
- Japanese New Year (January 1) - Eating traditional Japanese New Year's foods (osechi) and mochi (rice cakes) is big. Japanese people tend to send postcards for New Years instead of Christmas. Children receive their otoshidama, New Year's money packets. Poetry and games are also traditions, especially in the game of uta-karuta, where poems from the Hyakunin Isshu are used for a snap-style card game.
- Coming of Age Day (Second Monday of January, January 5 before 2000) - A day to congratulate and encourage people who have reached twenty years (the age of majority) after the previous Coming of Age Day. A kind of Coming of Age celebration.
- Setsubun (February 3) - While not a holiday, this day traditionally marks the beginning of spring. Throwing beans to ceremonially expel negative spirits from the household is a characteristic tradition. Technically, this is spring Setsubun, properly called Risshun; each seasonal division can be called Setsubun.
- National Foundation Day (February 11) - A celebration of the foundation of Japan and the imperial line by Emperor Jimmu in 660 BCE (Imperial Year 1 or Kōki 1).
- Valentine's Day (February 14) - Imported by chocolate companies, as a result of which the Japanese version of the holiday centers heavily around chocolate. Women give chocolates to their special men, as well as "obligation chocolate" to their male coworkers. The fact that only men get presents is (according to The Other Wiki) due to a typo made by an ad exec during the original advertising campaign. See White Day below.
- Parinirvana Day (February 15) - Not a public holiday. A Buddhist holiday celebrating the day when the Buddha and achieved Parinirvana, celebrated on the day of his death. It is a day for meditation and to reflect on one's own death and the departure of those recently deceased. Passages from the Mahaparinirvana Sutra are often read. It is called Nehan-e in Japanese.
- Doll Festival/Girls' Day (March 3) - Not a national holiday. A traditional day for young girls. Platforms covered in red carpet are set up with a display of dolls representing the Emperor, Empress and their court in Heian period dress along with palace accesories; It's complemented by Children's Day/Boys' Day (See Golden Week below).
- White Day (March 14) - Not a national holiday. On this day, men return the favour shown to them by women on Valentine's Day. Woe betide the anime man who forgets this! Like many "greeting card holidays" in the West, it was largely invented by marketers trying to give people an excuse to buy white chocolate.
- Vernal equinox (around March 20) - A day for celebration of spring and admiration of nature.
- Hanamatsuri (April 8) - Not a national holiday. Known as Buddha's Birthday in English, Vesākha in Pali, and a variety of names in Japanese, this is a Buddhist holiday celebrating the life of Gautama Buddha. In Japan, Buddhists pour ama-cha (hydrangea beverage) over small Buddha statues decorated with flowers. Formerly celebrated 8th day of the 4th month of the Chinese calendar before the Gregorian calendar adoption in the Meiji era.
- Golden Week (April 29, May 3-5) - April 29 is Shōwa Day, a day to reflect on the Shōwa era. May 3 is Constitution Memorial Day, a commemoration of the Constitution of Shōwa 21 (1946). May 4 is Greenery Day (April 29 before 2007), a nature celebration. May 5 is Children's Day (previously Boys' Day), which complements Girls' Day and is aimed at, well, young boys. All are days off. However, many Japanese take the entire week off, thus the name. A particularly important holiday in work-oriented Japan, and one featured in several anime. In practice, it's characterized by virtual gridlock in the otherwise excellent train system, prompting the media to breathlessly report on just how over-capacity the shinkansen is. Also the bane of manga fans, since  always takes the week off.
- Tanabata (July 7) "The Evening of The Seventh" - A holiday with romantic connotations. There is a legend in Asia called "The Weaver and the Cowherd" about two lovers (Orihime and Hikoboshi, who are the stars Vega and Altair) who can meet only once a year and only if it doesn't rain on the seventh day of the seventh month. (More information here on The Other Wiki.) It is marked with festivals and with writing wishes on strips of paper and attaching these to bamboo.
- Marine Day (Third Monday of July) - A day of thanksgiving for the ocean and its bounty which has sustained Japan through the ages.
- Bon (July or August 13-15) - A traditional Buddhist celebration aimed at alleviating the loneliness of deceased ancestors. It's often used as a day for family reunions and, since it's during the summer, it often involves outdoor festivals and the wearing of yukata. Buddhists visit and clean the graves of ancestors and the spirits of the dead revisit their household altars. Usually concludes with the floating of paper lanterns down rivers, symbolizing the return of the spirits back to the realm of the dead.
- Respect-for-the-Aged Day (Third Monday of September) - A day for veneration of elders and celebration of long life.
- Autumnal equinox (Around September 23) - A day to remember the deceased and one's ancestors.
- Tsukimi - Harvest moon festivals. The festival for the full moon is fifteenth day of the eigth month of the lunisolar calendar; the thirteenth day of the ninth month is for the waxing moon. These days are typically in September and October in the Gregorian calendar.
- Culture Day (November 3) - A day of promotion of culture celebrated typically with art exhibits and cultural festivals. It falls on the same day as the old holiday that celebrated the birthday of Emperor Meiji.
- Shichi-Go-San (November 15) - Not a national holiday, so it may be observed on the nearest weekend. It is a rite of passage celebration for seven and three year-old girls and five and three year-old boys, hence its name: Seven-Five-Three. Thousand year candy is given to children on this day.
- Health and Sports Day (Second Monday of October) - A day to celebrate sport and the health of mind and body.
- Halloween (October 31) - Not really celebrated; the Buddhist festival Bon may be the closest equivalent of what Halloween once was, since the spirits of the dead revisit household altars. Awareness of Halloween has increased of late as more and more American media makes it to Japan. Some anime use Halloween an an excuse to put a cast of cute girls in cute costumes. Generally there are a lot of (cute, not scary) Halloween decorations about for the month of October, but no one, or hardly anyone, actually dresses up or goes to parties.
- Labour Thanksgiving Day (November 23) - A celebration of labour and production equivalent to May Day or Labour Day celebrated elsewhere on May 1. (May Day is also celebrated by some trade unions in Japan).
- Bodhi Day (December 8) - Not a public holiday. A minor Buddhist holiday in Japan commemorating the Buddha's achievement of englightenment, like Vesākha/Hanamatsuri, which is a more widely celebrated occasion. Zen Buddhists call it Rohatsu; Tendai Buddhists call it Shaka-Jōdō-e. Traditions for its celebration vary between sects.
- Christmas Eve and Christmas (December 24-25) - In Japan, Christmas is considered a romantic holiday, to be spent with one's love rather than family (that is usually for New Years). This event is often a plotline in any anime featuring romance or romantic relationships. It is also celebrated in a more secular manner.
- Emperor's Birthday (December 23) - A celebration of the current emperor's birthday, which is twenty-third day of the twelfth month.
- Omisoka (December 31) - New Years Eve and the preparations in the week leading up to it are almost as important as New Year's Day itself and the period could be considered one "holiday" for the purposes of how it is portrayed. "Forget-the-year" parties are big at companies, and may involve drinking. A thorough house cleaning is often done. Special noodles are served on Omisoka itself. At midnight, a gong is rung 108 times.
Chinese New Year was once widely celebrated in Japan, but became rare after the Gregorian Calendar was adopted in 1873 to replace the Chinese-based lunisolar calendar. Chinese New Year may still be celebrated by Chinese and Koreans
and their descendents in Japan. Certain holidays and celebrations may be regional and not celebrated nationally such as those of Okinawa
For context, it should also be noted that certain days are not
holidays in Japan. Most notably, Saturdays are normally part of the established work week, though this has been changing in recent years. Sunday is not a universal day off, so shops and restaurants may have a closing day on any day of the week. Also, the school year is different, based on a school year beginning on April 1st and divided into trimesters separated by vacation periods. On average, Japanese students spend 60 more days in school than Americans, and these vacation times are valued highly. Up until 2002, Japanese students were required to go to school for a half day on Saturday in addition to the full days on Monday to Friday, and older anime may reflect this. However, a good thirty to forty of these days were for school festivals and cultural days. The eighth, fourteenth, fifteenth, twenty-third, and last two days of every lunisolar month are roku sainichi (six fasting days) or Uposatha, the Buddhist version of Sabbath.
For some excellent examples in anime, see Festival Episode
and School Festival