Check the airports on your itinerary
Jeju Island has a special visa-free entry scheme with nationality and itinerary conditions. Check those conditions before booking; an unsuitable route can prevent you from boarding. Korea's embassy notice sets the scheme out for "foreigners from visa-required countries who will be taking a direct flight from overseas to Jeju Island without stopping over at any other parts of Korea", and gives its own worked example: Singapore → Shanghai → Jeju qualifies, because the stop is in another country; a routing through Seoul or Busan does not, because arriving anywhere in Korea other than Jeju is exactly what the scheme does not cover.
So when you book, look at the airports on the ticket, not the airline. For this scheme, do not plan to arrive through another Korean airport before Jeju.
The special permission is limited to Jeju
The permission is geographic, and the notice says so in one sentence: "The visa-free entry scheme is applicable only to direct flights to Jeju Island. If you need to transit through other parts of Korea, you will need to apply for visa." That cuts on the way out as well as the way in. "A few days on Jeju, then a week in Seoul" is not a Jeju visa-free trip — you must meet the ordinary entry requirements for the mainland, including any visa your nationality requires, from a Korean embassy or the visa portal (“Do I need a visa to visit Korea?”).
Check the permitted length of stay with the Korean consulate or immigration before booking. Use the current guidance for your nationality and confirm the permission granted on entry.
Who is excluded
The scheme excludes some nationalities. Check the current list through a Korean embassy or immigration using the passport you intend to travel on. Do not rely on a copied list without checking whether it remains current.
Also check transit requirements in any country where you change planes. Eligibility for Jeju does not provide permission to enter or transit another country.
Where K-ETA fits
K-ETA means Korea Electronic Travel Authorization, a pre-travel approval for eligible visa-free visitors. If your passport is on Korea's visa-free list, check whether K-ETA or an exemption applies to your arrival (“Do I need K-ETA travel authorisation?”). If your passport needs a visa, K-ETA is not the route in; the Jeju scheme or a visa is. If you receive a visa, follow the stay and entry conditions attached to that visa rather than using the visa-free allowance.
