Tian
Tian can be translated into English as Heaven and within the context of Confucianism refers to the cosmic moral force and the highest order of the universe. Similar to the Dao/Tao in Taoism. For a Chinese emperor, after the Zhou Dynasty created the idea that there could only be a single legitimate ruler of China at a time, having the Mandate of Heaven (essentially the blessing of the gods/heaven) was crucial and illegitimately ruling without it caused unrest and rebellion, as it historically did.
Tian and humans
According to Confucius, the junzi (ideal person) would "stand in awe of Tian's ordinances", basically saying the noble person stands in harmony and follow Tian's guidance. Tian grants humans morality and cultivating said morality/virtue essentially equates to following Tian's will.
"Heaven"
Despite the translation, Tian should not automatically make you think of the pearly gates up above. Instead you should interpret it as an impersonal highest order of the universe, or simply just an invisible force, like gravity. Tian also doesn't perform miracles or grant anything but acts through (moral) causality, i.e. how something influencing another thing can lead to a certain outcome. Everything I've just said is generalized as other Chinese time periods interpreted Tian differently, which ties into the following section..
Interpretations by other dynasties
Interpretations by other dynasties varied radically, from concept to personal god to force.
The Shang Dynasty
The Shang (1600–1046 BCE) had an emperor that doubled as both a political and religious leader. They believed that their emperor was the only one capable of communicating between humanity and Shangdi (equated to Tian, though not a concept in this time + Shangdi was the family god of the Shang lineage), the supreme deity of all Shang rulers and capable of controlling harvests, war and various natural forces. The Shangdi was also prone to mood swings and often demanded sacrifices, which depending on if executed brought either fortune or disaster.
The Zhou Dynasty
The Zhou (1046–256 BCE) overthrew the Shang and introduced the Mandate of Heaven because they needed to replace the idea of the Shangdi granting legitimacy to the Shang dynasty, instead introducing the Tian, an impersonal force that granted rulers legitimacy only if they govern justly. "Heaven does not speak—yet the four seasons proceed, the hundred things arise." (Analects 17:19), unlike the Shangdi which was worshipped and prayed to.
Song & Ming Dynasties
Under this time period which lasted from 10th–17th c. CE Tian's definition started to change as Zhu Xi (Chinese philospher) called the Tian the moral and rational structure of the universe. The Tian was not a force with will, but rather an unfeeling order of everything. Later, according to Wang Yangming (also a Chinese philosopher), Tian's principles (morality) are instilled into the human mind at birth.
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Westerners, missonaries and Jesuits in particular sometimes translated Tian to the Christian God. Confucians today see Tian as more of a framework if anything, and Communist China (especially around founding) dismissed Tian as part of feudal society and nothing more than superstition.