Nora is the "doll" of the title. At the beginning of the play, she seems to be a simple character who is happy with her doll-like existence. However, as the play progressess, she realizes more and more the inequities of her lifestyle. We see the depth of her character develop before our eyes.
Nora plays her games with Torvald - dancing for him, playing the little squirrel, etc - in exchange for his taking care of her. The forged loan documents she has with Krogstad soon take away her carefree attitude, as she discovers that there is a limit to Torvald's sacrifices for her. This is shocking to her and starts her down the road to self-discovery.
She realizes that she was a doll in her father's house, taking on his beliefs, and that she now is a doll in Torvald's house, taking on his beliefs. She only plays with her children as dolls, and is raising them to behave the same. She recognizes that she must take the time to know and be her own self, in order to be the right sort of mother for her children, and thus leaves the house at the end of the play.
The door closing in the final moment of the play must have been enlightening for the audience (especially the women) in 1879, and still resonates today.