To put it simply, the Amgaeng-eosa was a type of official in the Joseon Dynasty whose job was to travel across the country and secretly inspect local magistrates to see if they were abusing the public.
They traveled incognito as it was feared if their identity was public, they would be a target for bribes. Going in disguise provided a way to approach the case with a fresh eye and no way for the target to influence his judge.
I guess the romantic image this creates is something like a cross between James Bond and Batman in ancient times. This position was pretty popular amongst the regular citizens and has resulted in a lot of folk stories, the most famous being Chunhyangjeon (춘향전) a love story in which an intelligent man falls in love with the beautiful daughter of a courtesan before he's forced to move away and follow his father.
The girl, Chunhyang, is made to tolerate the affections of a greedy lord in the meanwhile before being jailed after refusing him.
In the meanwhile, the boy (Mongyong) becomes a secret inspector, moves back to where Chunhyang is, punishes the lord, and lives happily ever after.
Not very original, but it's popular.
Did other nations have any similar systems in place? How effective were they? The time frame I had in mind really starts from anywhere up to the 1800's or so.
The obvious parallel is the Chinese Censorate (御史台). The hanja of Amhaeng-eosa is 暗行御史 and the position was part of the Samsa, the Korean Censorate (삼사, 三司). The Samsa shares the same Confucian philosophical foundations and origins as the Chinese Censorate. Like the Samsa, the Chinese censors were responsible for investigating all levels of the bureaucracy in search of corruption. They were active at the local level, acting as the "eyes and ears" of the government. The Vietnamese also adopted the Censorate system, which they called the Đô sát viện.