I can give you a general overview of this play, but also go to the links below to get more information. Enotes offers a summary, themes, and style, as well as essays and criticism of this play.
Cloud Nine has one major theme that enotes discusses. Churchill wanted to show the parallels between colonial power and sexual oppression. Both of them reflect the values of men. Clive represents the colonial attitude in the way he shows contempt for the African natives, but also in his remarks and treatment of Joshua, his black servant. Joshua has denied his heritage and only wants to be what the white men want him to be.
Betty, Clive's wife, Maud, and Ellen represent the sexism in the play through the dull, monotonous lives they lead. Their place is in the home, and their role as women is clearly defined. Like Joshua, Betty is also trying to be what the white men in her life want her to be.
The sexism of the men is shown in how they view all women as delicate, weak creatures to be protected. All of the men presume that the women should not and cannot even play catch.
Clive gets angry at his son, Edward, because he plays with a doll and isn't good at playing ball. We later find out he is gay.
The goal of this play is to break down the gender stereotypes. Act One is a satire of the Victorian family that represses the real feelings and desires of the characters. Act Two shows how the characters need
to be liberated from the constraints of their expected gender roles by showing that the family is no longer expected to be the traditional one.
Churchill throughout the play shows us how terrible colonialism is and how it can only end in resentment and violence.