The central conflict in Tennessee William's play A Streetcar Named Desire is between Blanche and Stanley. This conflict begins when Stanley notices that Blanche sees herself above everyone else.
The fact that Stanley is a strong man and Blanche is a weak woman immediately sets them at odds. Blanch sees herself above her sister's life and Stanley easily accepts his life in the flats in New Orleans. Given that Stanley recognizes the fact that Blanche looks down on him forces him to assert his masculinity over her.
Underneath, Blanche is a liar and Stanly is not. This, alone, causes conflict between them. Basically, Stanley has the guts to admit what he is and Blanche is constantly trying to hide who she is. This, also, sets up a conflict between Stanley and Blanche given his values conflict with hers. While readers, or watchers of the play, may not like Stanley, he is real. At no point in the entire play do readers see the real Blanche.
Both are strong in personality (in some aspects). This causes a problem between them based upon the fact that they are both trying to pull Stella in different directions.
In the end, their personalities are too similar in some ways and too different in others to allow them to get along. Neither looks at life in the same way and both have their way of looking at life in general. There are simply too many differences to allow them to get along. They were created to simply conflict with each other.
The basic conflict between Stanley and Blanche centers on the fact that Blanche moves in with them and won't move out because she has no place to go. There would be no conflict if Blanche had her own apartment and some source of income, but she has neither. She acts sweet and feminine, but she is desperate and fighting for her existence. She knows that Stanley wants to get rid of her by any means because she is a nuisance, an expense, and a threat his relationship with Stella. She obviously doesn't think Stanley is good enough for her sister and can't help trying to damage their marriage. In the meantime Stella is pregnant. The unborn baby poses an additional threat to Blanche because Stanley and Stella will need room for the baby which is being taken up by Blanche. The tininess of the apartment is part of the problem in this play. Even Blanches realizes this and is trying to solve her problem by finding a man to marry her. She is getting old. She can't find work as a teacher because she has a bad reputation for misconduct with teenage male students. She wouldn't consider a menial job such as doing housework. She dwells on the past and refuses to face reality. Stanley Kowalski is reality incarnate. The baby growing inside Stella's womb is reality incarnate. The terrible, cramped apartment is reality too.