Well in theory, Chinese dynasties typically followed what is known as the 嫡長子繼承制度 - male primogeniture. To give a little background, before the establishment of the Xia under Yu, all property was public and all leaders were elected. According to the Liji: [天下為公，選賢與能] - "All under Heaven is public, choose [those] who are worthy and capable." Yu was the first to transfer power to his son, thus beginning the rule of families (家天下). During the Shang, there was no clear rule on succession, and you will find that often times, younger brothers succeeded older brothers and there was a lot of strife over who should succeed. But toward the late Shang, distinctions began to made from the "eldest son born to principle wife" (嫡長子）and "common sons" (庶子) - those born to concubines. It's worth noting that an eldest son born to principle wife 嫡子 is not always an eldest son 長子. The Shiji notes that King Zhou of Shang Di Yi was born to his father's principles wife, but he had an older brother who was a "common son,” so Di Yi ended up becoming King even though he was not the oldest son. In other words, 嫡子 had to be born to the principle wife and always had precedence over his older brothers.

By the Zhou, succession from father to son was fully implemented and later codified into the Confucian classics. This was done for two reasons - one, to learn from the mistakes of the Shang and two, to solidify the Zhou royal family's control. From the Zhou on, it became standard (at least in theory) to pass the throne to the oldest son of the empress. However, as history would show, this was not always the case. Indeed, the question of "to set up [as Crown Prince] the eldest or the most worthy" (立長立賢) was always an issue which was a cause of concern for emperors. Emperor Wu of Han himself was the tenth son of Emperor Jing, born to a commoner who was a concubine. The first emperor of Tang Li Yuan was plagued by the issue of succession - he made his eldest son Li Jiancheng as Crown Prince because it followed Confucian protocol but he also wanted to make his second son Li Shimin, who contributed most to the Tang's founding, Crown Prince. In the end, Li Shimin killed his older brother during the Xuanwu Gate Incident and seized power, becoming Emperor Taizong of Tang. Taizong's successor Gaozong was likewise not the eldest child (though he was born to Empress Zhangsun).

Now the Ming and Qing is interesting because prior to the Ming, there was never any rules regarding the issue of succession. Zhu Yuanzhang was the first to implement such regulations. In his Instructions of the Ancestor of the August Ming (皇明祖訓), Zhu Yuanzhang wrote: [凡朝廷無皇子，必兄終弟及，須立嫡母所生者。庶母所生，雖長不得立] - "If the court has no princes, then the younger brother will surely [attempt to] succeed the oldest brother, so the son of the principle wife must be installed [as Crown Prince]. A son of a concubine, even if he is the eldest, shall not be installed." Under this principle, Zhu Yuanzhang passed over his other sons when his first Crown Prince Zhu Biao died and installed Zhu Biao's son (his grandson), as the next Crown Prince. He is Zhu Yunwen, or the Jianwen emperor, who was overthrown by his uncle Zhu Di. The Ming practiced male primogeniture, though there were instances where the emperor wanted to break convention. The Wanli emperor wanted to set up his son by his favorite concubine as Crown Prince, but the court unanimously opposed him and in a fit of rage, the emperor retired to the palace and refused to attend court for the next two decades.

The Qing was ruled by the Manchus, a semi-nomadic group from the northeast, so they didn't really follow the Chinese convention of successsion. Nurhaci's successor Hong Taiji was his eighth son. When Hong Taiji died without naming an heir, his brothers all started vying for the throne. A committe of Manchu princes chose his ninth son, Fulin, as emperor with Dorgon as regent. When Fulin (now the Shunzhi emperor) died of smallpox in 1661, his third son Xuanye was chosen to succeed him as the Kangxi emperor (possibly because he had already survived smallpox). Kangxi did appoint his son by his empress Yinreng as Crown Prince following Chinese customs, but subsequent events led to him stripping Yinreng of his position as Crown Prince. Kangxi's later reign was plagued by his sons vying for power, so the emperor announced that he would not publicly name another successor and placed his will inside a box behind the plaque in the Palace of Heavenly Purity, which was to be opened after his death. His successor was his fourth son, who is Emperor Yongzheng. To prevent strife over succession that he himself had faced, Yongzheng devised a new system where two scrolls containing the successor's name would be kept in two places. When the emperor died, the scrolls would be opened and if the names were identical, then that person would become emperor. From that point on, the Qing never installed a Crown Prince again and all successors were chosen after the emperor's death (with the exception of the Jiaqing emperor, because his father Qianlong abdicated).