The tragedy is the downfall and pitiable state of Blanche Dubois. Initially in the play, we see her as an interloper, interferring in Stanley and Stella's marriage and generally being a nuisance.
However, as the play progresses, she becomes more and more pathetic. We learn of her life as it faded from Southern Belle to near-prostitute, of her gay husband who killed himself after being discovered with another man. In desperation, Blanche tries to hold on to her illusions of youth and dignity, losing ground in every scene.
By the play's conclusion, Blanche has clearly lost all grip on reality. She is led away to an asylum, she makes a final grab at dignity, telling the doctor (who has just declined to put her in a straight-jacket) "I have always depended on the kindness of strangers."
It is not only a tragedy for Blanche. Stanley and Stella's world is forever shaken. More than ever before, Stanley realizes the divide in status between himself and his wife. Stella, for her part, has lost, probably forever, the sister she loves.