Red is generally called "aka （赤）" in Japanese. There are many traditional shades of red. The Japanese gave each shade of red its own elegant name in the old days. Shuiro (vermilion), akaneiro (madder red), enji (dark red), karakurenai (crimson) and hiiro (scarlet) are among of them.
The Japanese especially love the red that is obtained from safflower (benibana), and it was very popular in the Heian period (794-1185). Some of the beautiful clothing that were dyed with safflower red are well-preserved in the Shousouin at Todaiji Temple, more than 1200 years later. Safflower dyes were also used as lipstick and rouge by court ladies. At Horyuji Temple, the world's oldest wooden buildings, their walls were all painted with shuiiro (vermilion). Many torii (Shinto shrine archways) are also painted this color.
In some culture the color of the sun is considered yellow (or even other colors). However, most Japanese think that the sun is red. Children usually draw the sun as a big red circle. The Japanese national flag (kokki) has a red circle on a white background. Just like the British flag is called "the Union Jack," the Japanese flag is called "hinomaru （日の丸）." "Hinomaru" which literally means, "the sun's circle." Since "Nihon (Japan)" basically means, "Land of the rising sun," the red circle represents the sun.
There is a word called "hinomaru-bentou （日の丸弁当）." "Bentou" is a Japanese boxed lunch. It consisted of a bed of white rice with a red pickled plum (umeboshi) in the center. It was promoted as a simple, staple meal during World Wars, a time that was hard to get a variety of foods. The name came from the meal's appearance that closely resembled the "hinomaru." It is still quite popular today, though usually as a part of other dishes.
The combination of red and white (kouhaku) is a symbol for auspicious or happy occasions. The long curtains with red and white stripes are hung in wedding receptions. "Kouhaku manjuu (pairs of red and white steamed rice cakes with sweet beans fillings)" are often offered as gifts at weddings, graduations or other auspicious commemorative events. Red and white "mizuhiki (ceremonial paper strings)" are used as gift wrapping ornaments for weddings and other auspicious occasions. On the other hand, black (kuro) and white (shiro) are used for sad occasions. They are the usual colors of mourning.
"Sekihan （赤飯）" literally means, "red rice." It is also a dish that is served on auspicious occasions. The red color of the rice makes for a festive mood. The color is from red beans cooked with rice. A Baby is called "akachan （赤ちゃん）" or "akanbou （赤ん坊）." The word came from a baby's red face. "Aka-chouchin （赤提灯）" literally means, "red lantern." They refer to traditional bars that you can cheaply eat and drink at. They are usually located on the side streets in busy urban areas, and often have a red lantern lit out front.
Here are some other expressions including the word red.
The connotations of red in Japanese include "complete" or "clear" such expressions as "akahadaka （赤裸）," "aka no tanin （赤の他人）," "makkana uso （真っ赤なうそ）." The interesting thing is that it is black that has the connotation of completeness (e.g. a black lie) in English.