The people in the community do not have mirrors and do not approve of them.
Jonas comments that he does not really know what he looks like because the people in the community do not use mirrors. Mirrors highlight differences, so there would not be many of them around. Jonas’s comment about mirrors demonstrates the community’s official reason for not having mirrors: that they are not needed.
Mirrors were rare in the community; they weren't forbidden, but there was no real need of them, and Jonas had simply never bothered to look at himself very often even when he found himself in a location where a mirror existed. (Ch. 3)
However, if you look deeper at this “mirrors are not needed” explanation, you realize that the fact that mirrors are rare and Jonas never finds himself someplace where there is a mirror is not a coincidence. The minute Lily sees Gabe, the newchild their father brings home, she comments on his eyes. His light eyes are different, like Jonas’s. Jonas’s different eyes are a source of consternation for him.
The purpose of mirrors is for us to look at ourselves. We highlight differences in mirrors. We choose outfits, look at them in the mirror to see how they fit and look on us, and compare ourselves to others in a mirror. The community discourages and forbids all of these things. Everyone dresses alike. There are no colors. Differences are not mentioned, and are outright prevented as much as possible. When they do occur, like Jonas’s eyes and Fiona’s hair, they try not to talk about them.
When Jonas first notices colors, the thought of having choices occurs to him.
I was just thinking: what if we could hold up things that were bright red, or bright yellow, and he could choose? Instead of the Sameness." (Ch. 13)
At first, he is excited about choosing. Then it occurs to him that choice has been taken away because people might choose wrong.
This comment about mirrors is a telling one about the community. It is a sign that the community controls every little aspect of a person’s life, right down to their appearance. You do not even get to know what you look like, and you are not supposed to care.
Here is a short video that summarizes the analysis of The Giver: