The veil is quite the fashion statement, like wearing jeggings or a fedora. However, neither jeggings nor a fedora are closely tied with religious fundamentalism. The veil is. Although some women may choose to wear the veil, most women in Marji's family, including Marjane and her mother, do not. That should be fine—people should be able to wear what they please (although, we have to note, that most people do not look good in jeggings).
The Muslim regime of Iran thinks differently. They think that women should have to wear the veil, because everyone in Iran needs to follow their religion. Clothing is important to Iranians, because the smallest changes show your allegiance. Marjane tells us that "year by year, women were winning an eighth of an inch of hair and losing an eighth of an inch of veil" (34.11). Big battles are won by inches, not miles.
We're sure you've figured this out, but the veil is a big ol' symbol of how women are oppressed in Iran. Marjane has to put it back on before she can even enter the country. Wearing the veil is her least favorite thing about going back to Iran because it serves as a constant reminder that she is less of a person. We talk about women and oppression more in the "Themes" section, so if there isn't enough oppression in your life today, go check that out.