Titania's role in A Midsummer Night's Dream is to provide the conflict among the fairy world showing us that even the dreamlike fairy world is not perfect but also has its share of foolish behavior.
Titania's conflict with Oberon serves to portray a married couple who is on emotionally shaky ground. Also, Titania and Oberon are the only couple in the play who start out united, have a legitimate fight that is not magically induced, and then happily unite at the end. Shakespeare uses this couple to show us the foolish selfishness of love.
Titinia creates the conflict with Oberon by taking a particularly beautiful Indian boy that Oberon already wanted to train as one of his knights. Titania claims that she is refusing to give Oberon the boy because the boy's mother was her friend and died in child labor, so Titania promised to care for him, as we see in her lines:
But she being mortal, of that boy did die;
And for her sake do I rear up the boy;
And for her sake I will not part with him. (II.i.137-139)
Titania's refusal creates many fights between her and Oberon in which they even accuse each other of infidelity. Titania's fights with Oberon show us just how foolish lovers can be, especially with respect to selfishness. It further shows us that even the dreamlike state that the fairy world can create is not perfect. However, by the end of the play, Titania has relented and given up the boy. She has even made up with Oberon. Together, she, Oberon, and all the other fairies bless the couples sharing their wedding night in Theseus's home so that the couples will remain faithful and true to each other, as we see in Oberon's lines in the blessing,"So shall all the couples three / Ever true in loving be" (V.i.402-403). Her unity with Oberon and their joint venture to create everlasting love for the couples shows us Shakespeare's point that even though lovers act foolishly, once they set aside their foolish ways, their love can be mended.