There are two major conflicts in the story: Man versus Man and Man versus Himself.
Man versus Man is simply shown through the conflict between Erwin Martin and Ulgine Barrows; Martin, a quiet and reserved man, is shaken up and distressed by the brash mannerisms of Barrows.
The faults of the woman as a woman kept chattering on in his mind like an unruly witness. She had, for almost two years now, baited him. In the halls, in the elevator, even in his own office, into which she romped now and then like a circus horse, she was constantly shouting these silly questions at him. "Are you lifting the oxcart out of the ditch? ... Are you sitting in the catbird seat?"
(Thurber, "The Catbird Seat," jameshilton.com)
These mannerisms are annoying but not critical; Martin only buckles when he becomes convinced that Barrows is going to recommend the elimination of his own department. His original plan is something out of a murder mystery; although he does not intend it, his final solution is far more elegant. Barrows only realizes their conflict at the very end; she is entirely correct in her summation, but over-argued her case to the point where she was not believed.
Man versus Himself is shown throughout Martin's internal dialogue, which is shown as constantly going over his plan and second-guessing himself. He wonders if his choice in planted cigarettes is correct; he worries that someone will see him; he has an imagined deposition in court to justify his desire to murder Barrows. This comes to a head when he finds that he is incapable of following through with the murder, and despite the pressure and strain he formulates a new plan, which ends up working better than the original.
The only conflict in James Thurber's story "The Catbird Seat" is between Ulgine Barrows and Erwin Martin. She is a newcomer, but she has a special relationship with the owner Mr. Fitweiler, who is letting her reorganize the entire administrative part of the business. She has her eye on Martin's filing department, and he knows it is just a matter of time before this vulgar, inexperienced woman will want to wreak havoc with his carefully arranged files. She is the protagonist, and he is the antagonist. The conflict might be regarded as largely internal because she has not yet begun interfering with his department, although he knows it is only a matter of time and is frightened, worried, and anxious. It is possible that he could even think of murdering this woman if the situation became critical, but it has not reached that point yet. What he fears is that she will decide that the files are taking up too much space and that many of them are outdated, in which case she will want to have Mr. Fitweiler order Martin to throw half of them out, creating total havoc with his filing system. If the number of files is drastically reduced, it could mean layoffs in his department. It could even lead to his own dismissal.
What is important here is that Martin wants to nip the problem in the bud. If he waited until the axe had fallen on his department, it would be too late to commit a murder. He would be the prime suspect. The reader is actually led to believe that Martin plans to murder Ulgine Barrows on the night he visits her at her apartment. He is wearing gloves to avoid leaving fingerprints. He brings Camel cigarettes to provide a false clue. Evidently he has actually thought about committing a murder. If it occurs to the reader, it certainly must have occurred to Martin. It might be that he was even planning to commit a murder that night and changes his plan while he is in her apartment. He realizes that what he is doing is so out of character that no one would ever believe it. He exaggerates his impersonation of a wild, reckless, dissolute, potentially homiidal individual, and when Ulgine Barrows reports all this to Mr. Fitweiler the next morning, her employer, who has known Martin for many years, naturally thinks she must have been having hallucinations.
"The Catbird Seat" is a good introduction to the works of James Thurber, whose stories, essays, and cartoons were among the best things ever published in the New Yorker. Thurber and E. B. White were of the greatest importance in establishing the tone and high quality of that famous humor magazine. He writes about it in his book The Years with Ross. Harold Ross was the founding editor of the New Yorker and a very colorful and eccentric man who introduced many outstanding writers to the reading public.