Three ways to count age
International age (만 나이, man nai) is the age used in most of the world: you are 0 when you are born and add one year on each birthday.
Counting age (세는 나이) is the traditional Korean way: you are 1 when you are born and everyone adds a year on 1 January. A baby born on 30 December is already 2 on the next day.
Year age (연 나이) is the current year minus your birth year, without looking at your birthday.
Korea uses international age since 28 June 2023
Amendments to the Civil Act (민법) and the Framework Act on Public Administration (행정기본법) took effect on 28 June 2023. Since then, unless a law says otherwise, ages in administrative and civil matters — laws, contracts and official documents — are counted and written as international age. Source: Ministry of Justice press release (Korean).
Some laws keep their own rule. For example, the Military Service Act and the Juvenile Protection Act still count by year age. Source: Kyunghyang Shinmun Q&A, 28 June 2023 (Korean).
In daily conversation you may still hear people use counting age, so do not be surprised if a Korean friend says you are a year or two older than you think.