While not the catchiest of topics, knowing the difference between a gu, do, and gun can occasionally come in handy. Government designated divisions differ between the two Koreas, and for our purposes we'll just focus on South Korea. When the more general sounding term "Korea" is used, we are talking about South Korea.
Unlike Japan which uses prefectures, Korea is like China as the term "province" is used. Province is 도 (道, do) and the official names of provinces include a "-do" like Gyeonggi-do. Many people translate 도 to "province" like Gyeonggi Province, and sometimes just omit it all together, i.e., Gyeonggi. There are historically five provinces; Chungcheong, Gangwon, Gyeonggi, Gyeongsang, and Jeolla.
Today there are eight provinces as Chungcheong has been broken into North Chungcheong (Chungcheongbuk-do) and South Chungcheong (Chungcheongnam-do), Gyeongsang has been broken into North Gyeongsang (Gyeongsangbuk-do) and South Gyeongsang (Gyeongsangnam-do), and Jeolla has been broken into North Jeolla (Jeollabuk-do), South Jeolla (Jeollanam-do), and Jeju (Jeju Special Self-governing Province). As Jeju Island is a unique province with a separate police force and an independent visa policy that differs from the mainland, the term "special" (특별, 特別, teukbyeol) is in the title which leads some resources to state there are 8 provinces and 1 special province. When the peninsula was split along the 38th parallel, the provinces of Gyeonggi and Gangwon were split between the two Koreas.
Cities, counties, and districts are the three municipal-level divisions, but provinces cannot be subdivided into districts as districts are only within cities above a certain population. Cities are 시 (市, si) and often written as "-si" at the end of a city name; if used at all. Cities are defined as having a population of at least 150,000. Once a county has that many people, it converts into a city. If cities have 500,000 or more residents, the city can be subdivided into districts. More on districts is written below.
Seoul is the only "special city" (서울특별시, 서울特別市, Seoul Teukbyeol-si) as it is both the largest city and the capital. While it is in the middle of Gyeonggi-do, as of 1946 it technically is separate and doesn't belong to a province. The city name of Seoul has changed over different kingdom eras and the "special city" status was declared in 1949. Of the 51 million people who live in Korea, nearly 10 million of them officially live in Seoul. We say "officially" here as like other large cities, they keep growing which results in merging with or engulfing other cities, which makes it difficult to know where the lines should be drawn. If you count the cities the Seoul Subway serves, that would include Incheon City, Yongin City, and the rest of Gyeonggi-do. This larger megacity is sometimes called the Seoul Capital Area or Greater Seoul Metropolitan Area in which 25 million people reside; half the national population. Although not recommended, going from one end of the subway system to another end can take 2 1/2 to 3 1/2 hours.
Several presidential administrations have tried to spread government to different parts of the country, which eventually resulted in Sejong City being the one and only "special autonomous city" (세종특별자치시, 世宗特別自治市, Sejong Teukbyeoljachi-si). Sejong hosts a number of relocated government funded research centers and ministries, including our science-based parent ministry as of 2021. It is a bit difficult to say the number of residents as the city continues to expand. By the end of 2017 it was 280,000 and roughly 16 months later it was 320,000.
Cities with over a million residents are called metropolitan cities (광역시, 廣域市, gwangyeok-si) and must first be nominated before their name can change. Busan, Daegu, Incheon, Gwangju, Daejeon, and Ulsan have this title but Suwon, Changwon, and Goyang also have the residency requirement but have not been nominated. Metropolitan cities used to be called "directly-administered cities" (직할시, 直轄市, jikal-si) prior to 1995 as they, Seoul, and Sejong are run by mayors with four-year terms.
Other types of cities include specific cities, municipal cities, and administrative cities. For our purposes, we'll not go into those. When all classes of cities are taken into account, there are nearly 80 in the country.
If an area has a population under 150,000 it is called a county (군, 郡, gun). Officially these shouldn't exist in cities but exceptions can be found in Busan, Incheon, and Ulsan. While it is Romanized as gun, it has a different pronunciation that the weapon as the county gun is spoken with a closed throat. Like the -do and -si for province and city, respectively, gun is written as -gun. Not quite cities, counties are comparatively more rural meaning you won't come in contact with them very often. Gun are comparable to non-metropolitan districts in the U.K. and smaller than counties in the U.S. Counties can be subdivided into towns, and more on that below. There are roughly 80 counties in Korea.
If a city has a population over 500,000 the city can be subdivided into districts (구, 區, gu). Gu are similar to a borough or district in some Western countries. To make government offices more accessible in larger cities, most if not all paperwork you can do in a city hall (시청, 市廳, sicheong) can be done in a gu government office, neighborhood office, town office, township office, and some community centers. More on these other administrative divisions below.
Districts can be either autonomous (자치구, 自治區, jachigu) or non-autonomous (일반구, 一般區, ilbangu). The roughly 70 autonomous districts are in special or metropolitan cities which the 35 non-autonomous districts are in municipal cities. For all intensive purposes, they are the same.
Subdivisions
Districts (gu) and cities without districts are subdivided into "neighborhoods" (동, 洞, dong). A dong local government office is the smallest form of municipality to have their own office and staff. While there is legally one dong, it can be subdivided into several administrative dongs, each with their own office. An example of this is Myeongjang 1-dong (명장동1동, 鳴藏洞一洞, Myeongjang il-dong) in Busan. The name of a district and a neighborhood can be the same, for example 용산구 용산동2가 (龍山區 龍山洞二街, Yongsangu Yongsandong 2-ga) in central Seoul near Itaewon. The ga in that example is to split a dong with a large population for address purposes and does not mean a separate level of government exists. If you spend enough time in Seoul, you'll hear other ga instances and addresses. While dong can be split into tong (통, 統), this is rare in practice. Roughly 2,000 neighborhoods exist.
A subdivision of a county (gun) or a city with less than 500,000 residents is called an eup (읍, 邑). Frequently translated as "town", eups have a relatively large population of at least 20,000. You won't hear this term often as there are only a little more than 200 towns in Korea.
If there aren't enough people to form a town, a township (면, 面, myeon) is created instead. Consisting of at least 6,000 people, there are over a 1,000 townships but for obvious reasons they are only in more rural areas of counties or cities. Both towns and townships can be subdivided into ri (리, 里) which are confusingly translated as "villages". While this makes sense in a rural part of a province, this is odd when used within a city.
For additional information on the topic of Korean administrative divisions, you might want to check the Wikipedia page for South Korea or North Korea as they sometimes differ. Both of these articles contain links to other related pages.