The underlying theme is mendacity; the author develops this subject within the context of the family—the very social sphere that, more than any other, ought to foster mutual trust.
For years, the members of this family have acted in apparent harmony; when, however, it is revealed that Big Daddy, the family patriarch and “authoritarian master-father,” is suffering from cancer, the impending death overwhelms the protagonists: does truth emerge from this sort of catharsis, or a new lie? As Tennessee Williams explains in the “Notes for the Designer,” the setting is the bed-sitting-room of a great plantation house in the Mississippi Delta; two bachelors, the former owners of the plantation, once lived in this room, and therefore

the room must evoke some ghosts; it is gently and poetically haunted by a relationship that must have involved a tenderness which was not common

.
It is the appropriate setting for a play

that deals with the deepest extremities of human emotion, and needs that softness as a background

.
The theatrical work unfolds in three acts.
The protagonists of the first act are Brick, a former football player now turned alcoholic, and his wife, Margaret.
The man refuses to have any relations with his wife after she betrayed him with his best friend—who died, perhaps by suicide, in the wake of the adultery.
The loss of the only person who truly mattered to Brick has driven him to seek solace in alcohol.
While Brick pursues the victimhood of “the charm of the defeated,” Margaret refuses to surrender: she is keenly aware that the family is about to fall apart, and she does not want her husband to forfeit his social and economic standing to his brother.
Brick, deeply beloved by his parents, faces severe disadvantages regarding any inheritance: he is an alcoholic and, moreover, has not yet fathered any children, whereas his brother has six.
In her long conversation with her husband—in truth a soliloquy—Margaret wonders what the victory of “a cat on a hot tin roof” might be: “Just staying on it, I guess, as long as she can...” but then adds that silence does not work,

locking the door and shutting oneself in a burning house in the hope of forgetting that the house is on fire

.
In the second act, we witness the confrontation between Big Daddy and Brick, between father and son.
Here again, it is largely a soliloquy: Big Daddy does most of the talking, and we learn that this man—outwardly strong, resolute, and domineering—is consumed by disgust.

For all these goddam lies and liars that I’ve had to live with, and all this goddam hypocrisy I’ve lived with for all these forty years

; such is his disgust that at times he thinks it far better to have

a goddam blank (...) than all this crap that nature fills it with

.
Faced with this desperate confession, Brick replies that the conversation is

like all the others we’ve ever had together in our lives! It’s senseless, senseless, it’s, it’s painful (...) you don’t have a goddam thing to say to me!

Brick’s passivity suddenly turns into fury when his father voices the suspicion that there was something more intimate than mere friendship between Brick and his friend; and then Brick, cowardly like all the weak, reveals to his father the truth that had hitherto been concealed from him: he has cancer and is dying.
With Big Daddy’s desperate cry (“All liars, all liars...
lying, dying, liars!”) the second act comes to an end.
In the third act, the sons and daughters-in-law inform Big Daddy’s wife, who is also unaware, of her husband’s true condition; Brick’s brother attempts to persuade their mother to place all the estate in a trust, effectively cutting Brick out of the inheritance.
Brick remains as detached as ever, while Margaret fights desperately, to the point of telling a blatant lie: she declares she is pregnant and can therefore give Big Daddy a grandchild.
There are two versions of the third act.
In the original draft, the author imagines the story concluding with Margaret’s declaration of love to Brick, to which he replies with elusive cruelty: “Wouldn’t it be funny if that was true!”
In the second draft—the one staged, at the director’s behest—the author has Brick support his wife’s lie and accept Margaret’s unconditional love: “I admire you, Maggie” (says Brick), to which the woman replies:

what you need is someone to take care of you, gently, with love (...) and I can! I’m determined to do it, and nothing is more determined than a cat on a hot tin roof.

Has death finally dispelled the veil of mendacity? In the first version, Tennessee Williams answers in the negative: not even the “black thing” that comes “uninvited” frees us from hypocrisy and mutual distrust; human relationships are burdened by an existential condition of emptiness, clogged with too many lies—we are doomed not to communicate.
Certainly, this pessimistic vision cannot be dispelled by the conclusion proposed by the director, far too cloying and banal to ring true.
Tennessee Williams offers an observation particularly useful for understanding the play:

some mystery should be left in the revelation of character in a play, just as a great deal of mystery is always left in the revelation of character in life, even in one’s own character to himself

.
And thus we must look at how the characters evolve throughout the story, bearing in mind that “the actors must be allowed freedom on the stage.”
It is thus clear how Brick remains locked within his own world; Margaret’s personality, by contrast, emerges with increasing strength, as though she refuses to remain merely the wife of a broken man, standing ready instead to take the reins of the family.
Death reveals the true character of the human being.

The play is nearly flawless, which accounts for its great success.
While the third act, in its final version, is bustling and populated by a large cast, the first two acts suffer from a certain static quality, barely mitigated by offstage background noises and the fleeting appearances of minor characters.
On the page, the two acts come across as wordy and redundant; to be sure, that assessment might change under the light of theatrical performance.

Why read it? It is a great theatrical work, and the figures of Brick and Margaret possess tremendous fascination; to truly appreciate it, one must experience it in the theatre.

Cat on a Hot Tin Roof (La gatta sul tetto che scotta)