There is more than a single conflict in this story; however, the various conflicts all are centered around one main instigator. The conflicts are all triggered by Iran throwing off the previous modern cultural influences in favor adhering to a strict religious fundamentalist vision of Islam.
In 1979 a revolution took place. It was later called "The Islamic Revolution."
The story focuses on Marji who is the daughter of politically active parents that support Western values. Consequently, Marji begins life as a well educated, independent thinking girl. She is used to a modern lifestyle that includes playing cards, dancing, and political debates. Unfortunately, the religious fundamentalism that is returning works against all of those values. Marji is forced to go to an all girls school, wear a veil, and listen to the propaganda of her teachers. Satrapi writes, "We didn't really like to wear the veil, especially since we didn't understand why we had to."
Her Westernized family is persecuted, and some are even killed for their opinions and actions. Marji's conflict is an internal conflict about staying true to herself and her beliefs all while seeing friends and family members of hers arrested and killed for doing the very same thing. She recalls, "Between 1980 and 1983, the government had imprisoned and executed so many high-school and college students that we no longer dared to talk politics."
That clash of values is also an external conflict in the novel. Marji begins to actively rebel against the religious oppression by smoking, ditching school to flirt with boys, and listening to Western music that has been smuggled into the country.
Marji's vocal opposition to the new state continues to grow, and she continues to get into more and more trouble. Her parents are proud of her, but they fear for her life; therefore, the novel concludes when Ebi and Taji send Marji to a French school in Austria. They tell her, "'We feel it's better for you to be far away and happy than close by and miserable. Judging by the situation here, you'll be better off somewhere else.'"
I don't feel that this is a total resolution to the conflict. Iran is still struggling with the new fundamentalism that is there, and Marji is still strongly against it, but she isn't in the middle of it anymore. The conflict isn't so much resolved as it is put on hold until volume two of the story.