Starting from 2021.09.01(Wed), students from 112 countries that concluded a visa-waiver agreement with the Republic of Korea and designated visa-free entry countries must apply for K-ETA at least 24 hours prior to boarding the flight or ship bound for the Republic of Korea.
K-ETA is an electronic travel authorization that visa-free foreign visitors need to obtain before entering the Republic of Korea, by submitting relevant information online such as their travel information.

Thus, to apply for a visa after entering Korea as a visa waiver country citizen or a visa-free country citizen, you must apply for K-ETA before entering Korea.

1) How to apply: You can apply for K-ETA on the K-ETA official website (www.k-eta.go.kr) or mobile App (K-ETA).
    The applicant can receive the result by email, within 24 hours after completing the K-ETA application.

2) Period of validity: The validity period of a K-ETA is 2 years from the date of approval.

3) Fee: 10,000 KRW (additional fee excluded)

※ Unable to enter the Republic of Korea without K-ETA
※ During the pilot operation period (2021.05.03. - 2021.08.31.), you may enter Korea without having K-ETA, and the fee is waived.
※ Due to the COVID-19 situation, 91 countries/regions out of 112 countries/regions whose nationals are required for K-ETA have been suspended visa-free entry. 
    Therefore, as of May 2021, K-ETA will be implemented to nationals of 21 countries and regions that can enter Korea without a visa.

The following are the 21 K-ETA eligible countries and regions:
→ [ Albania, Andorra, Barbados, Dominica, Guam, Guyana, Holy See, Ireland, Malta, Mexico, Monaco, New Caledonia, Nicaragua, Palau, Saint Kitts And Nevis, Saint Vincent and the Grenadines, San Marino, Slovenia, United Kingdom, United States of America, Venezuela ]
※ K-ETA eligible countries are subject to change based on the COVID-19 situation. Please check the website announcements for details.

Please refer to the links below for more details:
1) Notice from Ministry of Justice  Information>What's new(immigration.go.kr)
2) K-ETA Application website  Korea Electronic Travel Authorization website > K-ETA
3) List of 112 countries (Q2)  Notice>FAQ>K-ETA