Learning about 자기 ( Oneself)

자기 is to demonstrate your idea of someone. To talk somebody else when you have a talk with other person.

그는 자기가 똑똑한줄 알아 = He thinks himself clever

그가 자기의 잘못을 인정했다 = He admitted that it was his fault.

I assume that you've heard of another word similar to this while watching korean dramas. 자신  is almost the same as 자기 

그는 자신이 똑똑한줄 알아 = Same as the above sentence
그는 자신의 잘못을 인정했다 = Same

However, there is another meaning of 자기. 자기 also means "Darling and Honey"
Therefore you need to be a bit careful when you use this word :D