Er, hello, it's the name of the book. Symbol alert.
The red tent is a place where the women can relax and get some distance from the rest of the world. It's a place where women bond, share stories, and bleed together. So when you boil it down to a symbol, it's a representation of femininity and female freedom.
Throughout the book, the inside of the red tent is almost like a secret world. Men aren't supposed to enter, and women go inside to do, well, stuff that women do—like menstruate. But there's something about the red tent itself that has an effect on the women: "The next morning, Leah and Rachel, side by side, walked out of the darkness of the red tent and back into the light of the world" (1.2.69).
Leah and Rachel fight often, but the red tent is a force that brings them together time after time. In the red tent, Leah and Rachel are sisters and friends. They share the same burden of menstruation. When they exit, they enter the real world—but this time, side by side.
The red tent is also the one place where women have power, since it's the one place they have to themselves. At this time, a woman's whole value was based on her body, her looks, and specifically her ability to bear children for her husband. The world of Genesis is a man's world. It's not the kind world where an independent woman could actually make it very far. So these women have to find power, friendship and independence where they can—in the women's world of the red tent.