I'm going to assume that you are talking about the movie, since All About Eve was based on a short story called "The Wisdom of Eve," and to my knowledge no book or stage play has been published.
All about Eve is the story of the novice, ladder-climbing Eve Harrington who befriends the famous Broadway star Margo Channing. Several other characters in the movie fall for Eve's naive persona: Bill Sampson (Director) and Lloyd Richards and his wife Karen fully believe that Eve is simply a smitten fan. Her charm convinces them to help her become successful. One character in the movie, Addison DeWitt (appropriately a critic), sees through her scam. Instead of exposing her, he blackmails her until Eve becomes little more than a puppet to Dewitt's wishes.
The movie All about Eve is told by Addison Dewitt, the critic, who is remembering the story of Margo Channing. In the movie, the young wide eyed Eve Harrington becomes Margo's secretary. Margo is a famous broadway actress who is overwhelmed with her life. Eve presents herself as a fan who worships and wishes to learn at the feet of her idol. Eve uses her position as the assistant to Margo to undermine the older actress in the eyes of her friends and her employers and to cast a favorable light upon herself. This includes convincing Karen to suggest her husband find a part for Karen in the play. Eventually Eve becomes Margo's understudy. When Margo fails to attend a reading of a new script, Eve reads for Margo and astounds everyone with her talent. She officially takes over the part when Margo suspiciously is unable to return for the evening performance because the car is out of gas. Dewitt seals Margo's fate by writing a glowing review of Eve.
Eve manipulates Karen and Richard until she becomes the new lead actress in Richard's new show. Margo is angry because of her perceived humiliation and betrayal by her close friends; she says she is not interested in playing the part anyway. Later on Dewitt comes to Eve and tells her that he knows all of her secrets. (Basically she has lied about everything including her name.) He informs her that he now owns her and her plan to marry Richard Lloyd will not happen and she will do everything he tells her to or he will expose her. She agrees because she has no choice.
The movie now switches back to the present where Eve is receiving a prestigious acting award. It looks as if she has managed to attain everything she ever wanted; however, she realizes she has no true friends. Her life is as fake as "Eve Harrington."
When she returns to her hotel room she meets a young actress named Phoebe. She is a fan and willing to help. Eve agrees. When Dewitt arrives at Eve's hotel room to give her the award she left on the table, he takes one look at Phoebe and sees immediately that the cycle has started again and Phoebe will be Eve's undoing just like Eve undermined Margo.