That was a long time ago, but it's wrong what they say about the past, I've learned, about how you can bury it. Because the past claws its way out.
-- Amir (Chapter 1 paragraph 1)

Importance: Amir is refering to the fact that he watched Hassan attacked and raped, and did nothing to stand up for him. This scars Amir, possibly more than Hassan, and he is never able to forgive himself for what he sees as an act of cowardice.

While my mother hemorrhaged to death during childbirth, Hassan lost his less than a week after he was born. Lost her to a fate most Afghans considered far worse than death: She ran off with a clan of traveling singers and dancers.
-- Amir (Chapter 2 paragraph 13)

Importance: Amir and Hassan are constantly compared, and this is one of many instances in which Hassan's life is considered worse than Amir's. However, Hassan's mother does return to him later in the story...