Almost all public transportation throughout the entire country can be paid with a single transit card (and you can even use them to pay at most convenience stores / vending machines). There are multiple transit card companies, but by far the most common is T-Money. In Seoul, T-Money can be purchased in almost every convenience store, often with various designs. There are also T-Money vending machines in the airport. Some stores throughout the city (such as ARTBOX) also sell unique T-Money cards with special designs.
Example T-Money Card
Vending Machine at Airport
Unique Designs (ex. from ARTBOX)
As mentioned in Can I use credit card in Korea?, you can buy a T-Money card with international credit card, but you can only add card balance with cash. See here for more information on getting cash in Korea.
Topping up the card balance with cash can be done with machines at any subway station, or with staff at any convenience store.
