Shosei means new emperor (mainly Crown Prince or Empress) starts to rule the government without having an enthronement ceremony after the former emperor died. There were two examples of Shosei in Japan, Prince Naka no Oe and Emperor Jito in Asuka period, both are recorded in the "Chronicles of Japan" as if they were practically real Emperors. In Japan, it is similar to Sessho (regency) although the Emperor also exist with the Sessho, the big difference between Sessho and Shosei is there is no Emperor in case of Shosei (the one who rules Shosei is practically the Emperor or a person who has political control equivalent to Emperor).
Shosei in China
The word, Shosei originally came from Chinese kanbun. (Sino-Japanese)
It is used when the Empress takes control of politics if the new emperor is too young to do so after the death of the former Emperor (Reign behind the curtain). It started in the Chunqiu era in olden times. In China the structure was different from Japan, there should have been the Empress who ruled the actual Shosei while the legitimate monarch (young emperor) existed at the same time.
In Japan there were two examples of Shosei which were, Prince Naka no Oe (Emperor Jomei and Empress Saimei's Prince, later called Emperor Tenchi or Tenji) and Emperor Jito (Emperor Tenchi's Princess, Emperor Tenmu's Empress and later called Emperor Jito), however there are some opinions that the three examples of Empress Jingu and Iitoyo no Himemiko (Princess Iitoyo) and Ahe no Himemiko (Princess Ahe) were considered as Shosei.
Prince Naka no Oe's Shosei
While Emperor Saimei died while raising an army against the Silla Kingdom, Prince Naka no Oe ruled the government when he was Crown Prince even though he did not become Emperor. Even after Emperor Saimei died, Prince Naka no Oe did not succeed to the throne, according to "Nihonshoki" (Chronicles of Japan) or "The Anthology of Myriad [Ten Thousand] Leaves," it is said there was a person called 'Nakatsusumera-mikoto' during this period, there was a strong theory that she was said to be Princess Hashihito (Emperor Kotoku's Empress). Since the Crown Prince had actual political control of the Japanese Shosei, it was different from the one in China where the Empress or the Empress Dowager used to take control of politics, it is presumed that above case was formally called 'Shosei' as the Empress Dowager (Emperor Saimei) was first appointed the nominal highest position followed by former Emperor's Empress (Emperor Kotoku's Empress, Princess Hashihito). In any case, Prince Naka no Oe moved the capital city to Imperial Palace Omi Otsu no Miya in 667, and he finally succeeded to the throne as Emperor Tenchi/Tenji on 3 January, 668.
(There is another theory that Prince Naka no Oe could not succeed to the throne fifteen years after Isshi-no-hen (a Murder in the Year of Isshi), since he had an affair with his real younger sister, Princess Hashihito, or as opposed to this, it was said Princess Hashihito (Nakatsusumera-mikoto) was enthroned as a temporal Emperor. There are various theories concerning the reason why Prince Naka no Oe was finally enthroned in his later years.)
Uno no Sarara no Himemiko's Shosei
After Emperor Tenchi/Tenji died, the Junshin War happened and Prince Oama (Emperor Tenmu) won the war, after he died as well, Uno no Sarara no Himemiko wished her son, Prince Kusakabe to succeed to the throne, however she put pressure on Prince Kusakabe's opposition, Prince Otsu to kill himself and was blamed for it, she was not able to make her wish happen (for Prince Kusakabe to succeed to the throne), also Prince Kusakabe had an ordinary character, (he was not good enough to become Emperor) thus the Empress Dowager, Uno no Sarara no Himemiko took control of politics to support Prince Kusakabe. Since Prince Kusakabe ended up as Crown Prince and died young, Uno no Sarara no Himemiko succeeded to the throne temporarily as Emperor Jito in the lunar New Year in 690, until Prince Kusakabe's son, Prince Karu (Emperor Monmu) became old enough to enthrone. Emperor Tenchi/Tenji's Prince, Uno no Sarara no Himemiko conducted the Shosei since she wished to secure her son, Prince Kusakabe to succeed the blood line to the throne. Although her son, Prince Kusakabe died young, she showed strong leadership and was brave enough to take control of politics to establish the Japanese nation under the ritsuryo codes, she finally passed the throne to her grandchild, Prince Karu (there was a theory Uno no and FUJIWARA no Fuhito planned an assassination when Prince Takechi, Emperor Tenmu's first Prince died just before Prince Karu's enthronement). The descendant of Prince Kusakabe, who was a child between Emperor Tenmu and Emperor Jito, became the main Imperial blood line during the Nara period and was actively involved in politics and culture, however the Imperial line was discontinued after Emperor Shotoku was the last Emperor, the Imperial line was returned to Emperor Tenchi/Tenji's line (Prince Shirakabe, later became Emperor Konin).
An example which had a possibility of being called Shosei
Empress Jingu's Regent
According to the "Nihonshoki" (Chronicles of Japan), after Emperor Chuai died, the successor to the Imperial throne, Emperor Ojin was just born, thus his mother, Empress Jungu became the regent.
In this case, it was called Regent in stead of being called 'Shosei.'
Of course this was an early Japanese legend and there was no record of Chinese words being used like Sessho (Regent) or Shosei, in the "Nihonshoki" it was called neither Sessho nor Shosei. However this raises a question as to why the description of Sessho and Shose were not unified in accordance with the government of Emperor Tenchi/Tenji's Shosei or Emperor Jito's Shosei when the Nihonshoki was edited.
The reason for above is unknown, although there are some possibilities of the following, 'since Emperor Chuai's death was not announced officially after he died, and it was temporary measure,' or 'there was no particular reason, the word Shosei was being used as a custom, since it also meant Regent after all,' or 'Shosei meant temporarily government since Emperor Ojin was officially considered as proper Emperor and he was called Emperor Taichu (Taichu means inside mother's womb).'
Iitoyo no Himemiko's (Crown Princess Iitoyo) government
According to the Nihonshoki, after Emperor Seinei died, since two Imperial successors, Emperor Ninken and Emperor Kenzo made mutual concessions and could not decide who to succeed to the throne, Itoyo no himemiko took control of the government, who was said to be their sister or aunt.
Ahe no Himemiko (Princess Ahe)
According to Shoku Nihongi (Chronicle of Japan Continued), after Emperor Monmu died, since the successor, Emperor Shomu was too young to enthrone and the issue as to who should succeed to the throne was raised, it was decided that Ahe no Himemiko rule the government temporarily, who was Emperor Monmu's mother and Emperor Tenchi/Tenji's Princess. However, the successor to the throne was not officially decided, and two months later, she was asked by the vassals and temporarily occupied the throne as Emperor Genmei until Obito no Miko (Prince Obito) become old enough to succeed to the throne. It was considered to be a period of two months, after Emperor Ahe no Himemiko died that she succeeded to the throne, and the official Shosei of Ahe no Himemiko was established, however, the government itself was not ruled properly during this short period of time.