A Streetcar Named Desire is one of the most important plays in American history. Today, the play has significance in many ways.
The first of these is that it defined an important genre in American theater. This genre is called "lyrical realism." Tennessee Williams pioneered the style of lyrical realism after World War II; previously, American theater had been defined by a style of gritty or dirty realism in the 1920s and 30s, which strove to accurately capture the struggles of working and middle class Americans. In Streetcar, some of Williams' characters use common dialect, and many of the scenes are sordid, involving street scenes and domestic violence. However, his stage directions--which call for soft, ethereal lighting--and Blanche's dialogue, which is poetic and romantic, attenuate this realism and create a layer of poetry that covers the play, as well. This was considered revolutionary at the time.
The second significance of Streetcar is historical. The play captures the transition of the American South from a planter aristocracy (represented by Blanche, an older, fading beauty who comes from a foreclosed plantation in the country), to an urbanizing, industrializing society (represented by Stanley, a Polish immigrant living in New Orleans and working in manufacturing). What is so interesting about Streetcar is that its attitude towards this transition is complicated: neither Blanche nor Stanley are represented as sympathetic characters. It still begs the question, how should we think about the past? What are the ethics of nostalgia? How do we compare the weight of injustices committed by societies (like Blanche's slave-owning ancestors) and by people (like Stanley, who commits rape)?