Koka means that an Imperial princess or a princess marries a man who is not a member of Imperial family nor the royal family. In Japan, koka refers to the case of marriage between a woman from Imperial family, especially an Imperial Princess, and a man from a non-imperial family (a subject).
According to "the Heir Decree" in the Ritsuryo codes, an Imperial Princess should marry an Emperor or a member of Imperial family who was within the fourth generations of an Emperor, and in ancient times there was no marriage between an Imperial Princess and non-imperial family. However, in Heian period, some Imperial princesses got married to subjects. This is koka, and it was extremely honorable for a man to take a Princess to his wife and there were many cases of koka in the middle of the Heian period. In "Genji Monogatari" (The Tale of Genji), a literary work of this period, we can find many cases of koka such as Omiya (the lawful wife of the Sadaijin [Minister of the left] and the mother of Aoi no ue [Lady Aoi]), Onna Sannomiya (the lawful wife of Hikaru Genji), Ochiba no miya (the lawful wife of Kashiwagi, who later remarried to Yugiri), and Onna Ninomiya (the lawful wife of Kaoru) and so on.
From the late Heian period to the Kamakura and the Muromachi period, koka of Imperial Princess came to an end because of the increase of nyoin (woman bestowed with the title "in") and the decrease of giving the title of Imperial Princess. In the Edo period, koka to Gosekke, the five biggest court noble families, started again. Later, in the end of Edo period, Imperial Princess Kazunomiya Chikako married to a Seii Taishogun (literally, "great general who subdues the barbarians"), Iemochi TOKUGAWA and this was the only case of koka to a samurai family.
Imperial Princesses kept their Imperial family status after they married to men from non-imperial families, so they weren't identified themselves by their husbands' family names. Therefore, such cases are not considered 'koka' in a strict sense. In modern age, however, Imperial House Act provides that an Imperial Princess should leave Imperial family when she marries a man from non-imperial family. Therefore Imperial Princesses who did koka (got married a man from non-imperial family) after the enthronement of Emperor Showa were identified themselves by their husbands' family names.
Imperial Princesses and Princesses who did koka
Although the marriage between MINAMOTO no Kiyohime, an Imperial princess of Emperor Saga, and FUJIWARA no Yoshifusa is regarded as the first case of koka, Kiyohime had already been given the name of Minamoto clan and descended to the subject status before their marriage, so the first case of marriage between an Imperial Princess and a subject is that of Imperial Princess Isoko (Kinshi), a daughter of Emperor Godaigo, and FUJIWARA no Morosuke. However, Imperial Princess Isoko supposedly did not get permission from her father Emperor and therefore it is said that the first case of koka with an emperor's approval is that of Imperial Princess Teishi, a daughter of Emperor Sanjo.