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These are RFID cards (sometimes referred to as NFC, although technically the two are different). They consist of two parts - an interrogator (the reader) and the tagged device (in this case, the card).
The same technology is used in Dubai for the public transportation system; and is available in most mobile phones.
You get a special SIM which is encoded with a NFC tag; you still need NFC technology on your phone; with the exception of iPhones (which have NFC technology, but is only accessible to Apply Pay); most modern phones are compatible.
Here is a page on du, a mobile operator in the UAE that has some more details.
For the credit/debit cards, it is the same idea - except the tag is embedded in the debit/credit card. Your card must support this; and the technology is marketed as MasterCard Contactless (used to be called Paypass), Visa calls it payWave, and American Express calls it ExpressPay.
Specifically for T-money it is an embedded RFID device, which is also available as a SIM card and a phone application. However, you cannot use your credit card as t-money cards; you can get a refillable t-money card (its called the City Pass Plus Card), and you can reload it via a debit/credit card.
Just like the normal (non-reloadable) t-money cards, this one also never expires you can buy one and use it for your subsequent visits.
You can find out more information on t-money at this website.