"When you call someone, you either call him by his first name or using his/her family name with Mr. Mrs. or Miss before it. According to your knowledge of that person.
In Iraq it’s a little different. You call him/her with the first name but after getting married and have a child you call him with his/her son/daughter name but you put the word Abu which means ‘Father of’ like Abu Ali means father of Ali and if he has a daughter you call him Abu Zainab for example. For mothers you put the word Um before the name of the child which means ‘mother of’. It’s done to show more respect than using his/her first name especially for old people. I know many people by their Abu or Um and I don’t know their real names, because they don’t use them any more."
Anthropologists call this teknonymy, which means "naming after one's child." It's a custom that tends to de-emphasize ancestry, as compared to our American use of surnames that emphasize patrilineal ancestry.
A discussion of naming practices in different places may be found here.