"Свой" is used instead of my, your, his etc. when it refers to something belonged to the subject of the sentence. Literally it can be translated as "one's".

I took my bag. Я взял свою сумку.
He took my bag. Он взял мою сумку.
He took his bag. Он взял свою сумку.

AFAIK using мой, твой, его и т.д. instead of свой is not considered a formal error in Russian but native speakers would hardly say something like that.

Дайте, пожалуйста, мою сумку! - Give me please my bag!
Дайте, пожалуйста, свою сумку! - (You) Give me please your bag!