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

The adjective “자랑스럽다” (ja-rang-seu-reop-da) means “to be proud.” It describes the feeling of pride or admiration toward oneself, others, or accomplishments.

Breaking It Down:
자랑 (ja-rang) – Pride / Boast
스럽다 (seu-reop-da) – To be (used to form adjectives)

Example Sentences:

너무 자랑스러워요! (neo-mu ja-rang-seu-reo-wo-yo!) – I’m so proud!
부모님이 자랑스러워하셨어요. (bu-mo-nim-i ja-rang-seu-reo-wo-ha-syeoss-eo-yo.) – My parents were proud.
우리 팀이 이겨서 정말 자랑스러워요. (u-ri tim-i i-gyeo-seo jeong-mal ja-rang-seu-reo-wo-yo.) – I’m really proud that our team won.
자신의 노력에 대해 자랑스러워하세요. (ja-sin-eui no-ryeok-e dae-hae ja-rang-seu-reo-wo-ha-se-yo.) – Be proud of your own efforts.
한국 문화를 알릴 수 있어서 자랑스러워요. (han-guk mun-hwa-reul al-lil su iss-eo-seo ja-rang-seu-reo-wo-yo.) – I’m proud to be able to share Korean culture.

자식 literally means “child” or “offspring.”
It refers to one’s own children (sons and daughters).

Depending on tone/context, it can be affectionate, neutral, or even slightly derogatory.

Affectionate: my beloved child
Neutral: offspring / children
Derogatory: that brat / that kid (when used in anger)

Example Sentences

그는 두 자식이 있어요. → He has two children.
자식을 위해서라면 뭐든 할 수 있다. → I can do anything for my children.
저 자식 때문에 일이 망쳤어. → Because of that brat, things got ruined. (Here, 자식 is used negatively)
부모는 자식을 끝까지 사랑한다. → Parents love their children until the end.

Idioms & Figurative Uses with 자식

자식 걱정은 끝이 없다. → Worrying about children never ends. (Common saying among parents)
자식 이기는 부모 없다. → No parent can win against their child. (Meaning: parents always give in to their children’s wishes)
자식은 부모의 거울이다. → Children are the mirror of their parents. (Children reflect their parents’ behavior/values)
자식은 빚이다. → Children are like a debt. (Figurative: raising children requires endless sacrifice and responsibility)

Nuance

When parents talk about their kids, 자식 is warm and affectionate.
When used in anger (like “저 자식”), it can mean “that guy / that brat” — not necessarily about one’s own child.
So context is everything: 자식 can shift from tender to scolding depending on tone.

자신 (jasin) has two main meanings > self/oneself, and confidence. Depending on the person used in the sentence, it could mean oneself, myself, yourself, himself, herself, themselves

자신 has a meaning very similar to 자기.

“Oneself” or “Self” (自身)
Refers to a person themselves, emphasizing individuality or self-reliance.

자신을 믿어야 한다. → “You must believe in yourself.”

“Confidence” (自信)
Describes a strong belief in one’s abilities or success.

그는 자신이 있다. → “He has confidence.”

If a person is mentioned twice in the same sentence, but not both are used as subject, then 자기 is used.
If however both times the person is mentioned as subject, then 사진 is used and 사진 is used as pronomen.

The standard translation is oneself, but depending on the person the translation could then change to myself, yourself, himself, herself, themselves

저는 자신을 안 믿어요 = I don’t trust myself
너는 자신을 안 믿어? = You don’t trust yourself?
그는 자신을 안 믿어요 = He doesn’t trust himself
그녀는 자신을 안 믿어요 = She doesn’t trust herself
그들은 자신을 안 믿어요 = They don’t trust themselves

It is common to place the following words before 자신:
나/내/저/제 if the subject is “I”
너/네 if the subject is “you”
그/그녀/자기 if the subject is “him” or “her”
그들 if the subject is “they”

For example:

저는 제 자신을 안 믿어요 = I don’t trust myself
너는 네 자신을 안 믿어? = You don’t trust yourself?
그는 자기 자신을 안 믿어요 = He doesn’t trust himself
그녀는 자기 자신을 안 믿어요 = She doesn’t trust herself
그들은 그들 자신을 안 믿어요 = They don’t trust themselves

Sometimes the person is the subject and also placed before ~에 대해 or ~에게 in a sentence.

For example:

… 자신에 대해 … = about oneself
… 자신에게 … = to oneself

Below are many examples showing the use of 자신:

청소년들은 그들 자신을 사랑하지 않아요
= Young people don’t love themselves
그 학생은 자기 자신에게 자꾸 변명을 해요
= That student always makes excuses to/for himself
저도 제 자신을 몰라요
= Even I don’t know myself
저는 제 자신에게 그런 말을 많이 했어요
= I said that (type of thing) to myself many times
그 남자가 수업 시간 동안 자기 자신에게 말을 해요
= That boy talks to himself during class
저는 한국에서 제 자신에 대해 많이 배웠어요
= I learned a lot about myself in Korea
그는 자기 자신에 대해 아무 것도 알고 있지 않아요
= He doesn’t know anything about himself

see also here

“자신감 있다” (jasingam itda) means “to have confidence” or “to be confident.” It describes someone who believes in their abilities and approaches situations with assurance.

Here are some examples:

그는 발표할 때 항상 자신감 있다.
(Geuneun balpyohal ttae hangsang jasingam itda.)
→ “He is always confident when giving presentations.”
자신감 있는 태도가 중요해요.
(Jasingam inneun taedoga jungyohayo.)
→ “A confident attitude is important.”
그녀는 자신감 있게 대답했다.
(Geunyeoneun jasingam itge daedaphaetda.)
→ “She answered confidently.”