자기 (jagi) is a versatile Korean word with multiple meanings depending on the context: self/oneself, Darling, magnetism
자기 has a meaning very similar to 자신.
“Oneself” or “Self” (自己)
Used to refer to oneself or someone previously mentioned.
자기가 결정해야 해. → “You have to decide for yourself.”
“Honey” or “Darling” (Endearing term)
Often used between couples as a term of affection.
자기야, 뭐 먹을래? → “Honey, what do you want to eat?”
“Magnetism” (磁氣)
Refers to the physical property of magnets.
자기장이 강한 곳이다. → “This place has a strong magnetic field.”
“자기소개 좀 해봐” = “Tell me about yourself.”
“(자기) 자신을 더 믿어봐” = “Believe in yourself more”
When referring to somebody in the third person, you can use words like “he,” “she,” or “that person.” For example:
그는 한국을 좋아해요 = He likes Korea
그녀는 한국을 좋아해요 = She likes Korea
그 사람은 한국을 좋아해요 = That person likes Korea
자기 is usually used in sentences when a person has already been mentioned, and is being mentioned again.
“He likes his appearance.”
You could write that sentence like this:
그는 그의 외모를 좋아해요
However, it could also be written like this:
그는 자기(의) 외모를 좋아해요 = He likes his appearance
Here, you can use 자기 to replace the person you are talking about the second time you mention that person.
Examples:
우리 아들은 자기(의) 일을 항상 혼자 하고 싶어요
= Our son always wants to do his work alone
저의 아내는 자기가 요리한 것을 보통 안 먹어요
= My wife usually doesn’t eat the food she cooks
우리 아들은 자기 건강에 대해 의사와 상담을 했어요
= Our son consulted with a doctor about his health
여자가 변한 자기 모습을 보고 실망했어요
= The girl looked at her changed appearance and was disappointed
In the example sentences above using “자기,” some person is being referred to twice in the same sentence. Although this person is mentioned twice, the person is not the subject and the object of the sentence ant the same time.
For example:
우리 아들은 자기(의) 일을 항상 혼자 하고 싶어요
The subject is “our son.” The object is “his work.”
저의 아내는 자기가 요리한 것을 보통 안 먹어요
The subject is “my wife.” The object is “the food (she cooks).”
When one person is both the subject and the object of a sentence, the word “자신” is used as the object. It is used like a pronoun.
see also here