An introductory reading about the Joseon state,^(1) mentions that the censurate was a powerful branch of the government that made politics more highly visible. I've never heard of an institution like that, even in other Neo-Confucian states in East Asia. What were they and what did they do?
“Creating a Society of Civil Culture: The Early Joseon, 1392-1592,” by Jaehyun Kim Haboush, p 3-13
I can't really say much about the Joseon Censorate, but given how closely Korea modeled its governmental institutions off of Ming China, I think I can give you a good idea of how Joseon's censorate operated by talking broadly about the imperial Chinese censorate.
For about two thousand years, the imperial Chinese state maintained a censorate of one sort or another and the final two imperial dynasties, the Ming and the Qing, were no different. By the late Ming, the ministry was called the "都察院" (duchayuan) and was extremely powerful, having been raised to the level of the 6 traditional ministries proper which formed the highest level of Chinese civil administration (as well as the Joseon government, although the Chinese and Korean ministries had slightly different names; the Ming referred to its ministries as 部 whereas Korea used 曹 so the ministry of finance in the Ming dynasty was the 户部 vice 户曹 in Korea).^(1) The reason for the high rank of the Censorate was because the agency served as the overall investigative and governmental accountability institution in Imperial China and in countries which modeled their governments off of China (namely Korea and Vietnam). In China, the agency gained the nickname "天子耳目", or "Eyes and Ears of the Emperor". By the end of the Ming, the censorate was nominally led by two very senior officials, dubbed the Censors-in-Chief (都御史), each holding rank two-primary (to put things in perspective, the Chancellor in Ming China was only one rank above the Chief Censors and the Chief Censors themselves were at least as high-ranking as the ministerial leaders). Under them, numerous other officials holding various ranks staffed the Censorate, from Vice Censors-in-Chief to investigative censors. Investigative censors did the majority of the grunt work for the agency and as a whole, they were perhaps the most feared among the thousands of middle-ranking officials in the empire. It was up to the investigating censors to travel far and wide throughout the country to monitor other officials of ranks both low and high and ensure that they were performing their duties honorably. Instances of impropriety could result in reports, admonishments from the capital, or more severe punitive actions being taken against the offending official. Regional Inspectors, who were in charge of the Censorate's activities in a particular region (what a surprise) wielded even more power, sometimes having a direct line to the Emperor himself. These men could order audits of local records, interrogate officials, and even remove officials from office if he saw fit (this had to be justified with a memorial sent to the throne, but it was possible). That was the general purpose of the Censorate. It can be essentially understood as a sort of proto-governmental accountability watchdog, sort of like an early modern version of the United States' GOA.
Now for a bit of a history tangent about the Chinese Censorate in particular.
In the early days of the Ming, there was a little bit of messiness when it came to the overall organization of the Censorate, driven in part by the paranoia and ruthless autocracy of the first Ming emperor, Ming Taizu. In the 1380s, Ming Taizu essentially decapitated the main governmental ministries as a result of some rumors regarding possible treason but the Censorate was nominally reconstituted into its original form. However, following this reshuffling, the Chief Censors saw a significant decrease in the amount of power they wielded over the Censorate as a whole, and it seems that the Emperor wrested control of investigative censors and regional inspectors from the agency itself and put them under his direct control. Nonetheless, the purpose of the agency remained the same - monitoring the performance of the imperial officials and punishing them for their misdeeds.^(2) Later emperors would also change the degree to which the Censorate was an 'independent' arm of the government, from the tight control of the first emperor, to the very lax control exercised by the longest reigning Ming emperor, Ming Shenzong/the Wanli Emperor.
Towards the end of the Ming dynasty, the Censorate was not above getting entangled in the messy factionalism and intrigue that mired the final Ming emperors, starting with the end of the 16th century. As the various ministries and senior officials in the Wanli Emperor's administration bickered amongst themselves, there was a corrupt alliance of some parts of the government with the Censorate. This reduced the Censorate's effectiveness greatly and enabled a culture of corruption that permeated the upper echelons of government. Silver from the treasury was redirected to allies of the Censorate, officials who were endlessly corrupt were given free reign, and those with the right connections could find themselves appointed to high offices for which they were ill-suited. At this point, the actual efficacy of the Censorate had completely broken down, and the pervasiveness of governmental decay would eventually help accelerate the ultimate collapse of the Ming dynasty in 1644.^(3)
Sorry I can't help you with the specifics of Korea's Censorate as I'm not very well-versed in Korean history on its own but I hope this can at least give you a starting point.
Additional Reading about the Chinese Censorate if interested (a bit dated but the overall historical summary is still not bad): https://www.jstor.org/stable/1951246