Literally, the title, A Streetcar Named Desire, refers to the streetcar Blanche rides into town on. Of course, desire plays a much larger role in the play than that.
Desire is why Stella stays with Stanley. After he abuses her she looks forward to the passion they'll share when they make up. Desire is what Stanley can fulfill for her that her former plantation life, Blanche, etc., cannot. Desire is connected to violence in the play, and is prevelant. In addition to Stanley and Stella, the couple who live above them also have a relationship that contains violence.
Desire also plays a major role in Blanche's character. It's part of the reason for her former life that she tries to hide from Stanley and Stella, etc., it is revealed when she attempts to seduce the young boy that enters the house, and is revealed by her flirting with Stanley. The play reveals a disparity between how women act in Southern society, and how women are expected to act in Southern society. Blanche is figuratively trapped between how she wants to live, and how she is forced to live.
Violence, of course, mixed with desire--Stanley's--also destroys Blanche and leads to her madness.
The play basically breaks down into a battle for Stella between Blanche and Stanley. And Stanley wins, largely because he fulfills Stella's desire.