Fèng guān凤冠(Phoenix Crown) of Chinese empresses, Qing dynasty and Ming dynasty, collected in the Forbidden city, Beijing. Actually the first red crown is for high-ranked imperial concubines whose title is fēi妃, and people call them huáng fēi皇妃. The second red crown with more gold ornaments is for the empress who is the only wife of emperor by ancient law, her title is hòu后 and is always called huáng hòu皇后 by people. Huáng皇 means imperial/royal.
But pay attention that the title fēi妃 also applies to the wife of prince because her status is inferior to the status of empress/huáng hòu皇后, for example, the wife of prince is called wáng fēi王妃, and if the prince is tài zǐ太子(crown prince), then his wife is tài zǐ fēi太子妃.
In English the wife of prince can be called princess, same as the daughter of emperor. However in Chinese the wife of prince and the daughter of emperor don’t share the same title. Usually the daughter of emperor is considered more superior than the wife of prince in ancient China, for the latter probably doesn’t own royal blood. The daughter of emperor is called huáng nǚ皇女 which straightly means ‘imperial/royal daughter’. If she is especially in favor with the emperor or her mother is the empress, then she would be given the distinguished title gōng zhǔ公主 by emperor himself, so not every huáng nǚ皇女 can be called gōng zhǔ公主. There are even variations in the title gōng zhǔ公主, such as dà gōng zhǔ大公主(most noblest daughter of emperor), zhǎng gōng zhǔ长公主(most noblest sister of emperor), dà zhǎng gōng zhǔ大长公主(most noblest sister of the father of emperor, his aunt).
Western terminology is often insufficient for translating East Asian royal titles! I usually translate 长公主 as Princess Royale and 大长公主 as Grand Princess. In Korea, where titles are a rank lower than those in China, the Princesses are called 翁主, which I translate into Grand Duchess. Japanese titles don’t have this problem, as their princesses are called Uchinomiko or Himemiko
All daughters of the Emperor would receive titles by the time they were wed. However, the parents’ favor would decide whether or not they received titles before that. Only an enfeoffed princess is eligible to be referred to as Princess and granted a title to go along with her fief. The process would be similar for princes, though on average, they were granted fiefs much earlier than their sisters. In this case, evidence of the parents’ favor would be a later investiture time, as a titled prince was required to go rule over his fief, which would be miles away, essentially cutting him off from his parents. So less favored princes would be sent packing as soon as they could walk and read. Other favored ones stayed until they were 12 or 13. In some rare cases, they stayed well into maturity.
Phoenix crowns are actually a rather recent part of the Imperial dress code, officially being included only about a thousand years ago. Prior to the North and South Dynasties, royal ladies wore an assortment of ornaments, including a buyao which featured flower clusters and phoenixes. After the North and South Dynasties, royal ladies wore flower crowns. In both cases, rank was decided by the clusters of flowers. Phoenix crowns were actually considered formal, but not ceremonial wear prior to the Song Dynasty. During the Song Dynasty, royal ladies spiced up their flower crowns by including trendy ornaments like phoenixes, so by the Yuan and Ming Dynasties, the phoenix had become the main feature. Technically, only the Empress had a Phoenix Crown. Imperial consorts had Bluebird Crowns, and everyone else had a Pheasant Crown. However they all look so similar that they get lumped together as Phoenix crowns.