A Basic History
Construction of Seoul's first subway line started in 1971. It officially opened to the public on August 15, 1974, a 7.8 km (4.8 mi) line from Cheongnyangni Station (University of Seoul) to Seoul Station. The line connected to larger Korail (the national rail operator) regional services expanding beyond the city limits, similar to the RER in Paris. The underground section was operated by Seoul Metro, a company owned by the Seoul Metropolitan Government. This set the basis for a system that today spans most of the city limits and beyond, and is one of the most efficient in the world.
By Thyristorchopper - 자작, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=12183290
The most recent video I could find showing a timelapse of the metro's construction. Unfortunately there are no English captions.
The system does not operate 24/7. While operating hours vary per line, the system generally operates from 5:30 am to 1:00 am on weekdays, and from 5:30 am to midnight on weekends.
WiFi and internet service is installed on every train and in every station. Almost all stations feature screen doors.
Fares
T-Money, one of the primary handlers of fares in Seoul.
The system operates on a unified transportation system, meaning that subways and buses in Seoul, Incheon, and the province of Gyeonggi-do are treated as one unified system when it comes to fares. With the exception of the Shinbundang Line, EverLine, and the U Line, every subway line can transfer to the other for free. All city buses have free transfers.
The current single-use ticket is a plastic card with RFID technology. They are available from automated machines at every subway station. A 500 won (41 cents) deposit fee is included in the price, and is refunded when the ticket is returned at any station.
Multiple-use cards can be sold at many convenience stores, and the function is available in many credit/debit cards.
Current fares are 1250 won (~$1) for a trip up to 10 km. For every subsequent 5 km, an additional 100 won fee is added (~8 cents). Once 50 km has been passed, 100 won is added every 8 km.
Children can ride for half-price. Senior citizens (over 65) and disabled people can ride for free.
The two primary operators of the system are Seoul Metro and Korail. Seoul Metro and the SMRT (Seoul Metropolitan Rapid Transit Corporation), which operated lines 5-8, merged in 2017.
With a few exceptions, numbered lines are considered urban rapid transit lines within the Seoul National Capital Area (SNCA). Named lines operated by Korail provide a metro-like commuter service that usually extends well beyond the boundaries of the SNCA.
You can read more about each individual line by clicking its icon below.
Main metro lines
Commuter/named lines
Light rail lines
Left or Right Hand?
When Korea was under Japanese rule, both roads and rail tracks were built for left-hand traffic. After the peninsula was liberated at the end of WWII, all roads were converted to right-hand traffic. However, the rail lines were not converted: the extent of the changes, which included switch conversions and edits of destination signs, would be a lengthy, expensive process. So, the policy for left-hand traffic remained with rail lines.
Things began to change with the time of the subway's opening. Official policy was updated, now stating that subsequent lines classified as "subway lines" would be built for right hand traffic. Lines classified as "commuter and regional" lines would remain at left-hand traffic. Line 1's case was rather unique because the subway would have to link up with existing left-handed commuter lines. So, the subway, and subsequently the rest of what makes up the current Line 1, was built on the left-hand side. All subsequent numbered metro lines were built on the right hand side.
A comparison. What is "forward" on Line 1 is "backward" on Line 7.
Top image by Mtattrain - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=46751591
Bottom image by Minseong Kim - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=77052015
Because of the junction's likeness to twisted doughnuts (꽈배기), junctions of this type have been called 꽈배기굴, roughly translated to "twisted doughnut junction."
Back to Seoul Metro
To Trains
This new policy, however, brought some problems in the future. In the early 1990s, plans were being drawn up to extend Line 4 all the way to the Ansan Line, called the Gwacheon Line. Line 4 up until that point had been built on the right-hand side, as per policy. However, the Ansan Line was still considered a commuter line (it would not become part of Line 1 until 2000, and Line 4 until 2003), and had been built on the left-hand side.
After negotiations between Korail and Seoul Metro on whether to make the entire line on one side came to an impasse, a compromise was reached. The Gwacheon Line would be built on the left-hand side, with a junction between Namtaeryeong (the last station under Seoul Metro control) and Seonbawi (the first station under Korail control) allowing trains to cross over from one side of the track to the other.