There are several different conflicts present in this story. The first and most pronounced is the human v. human conflict between Ms. Washington and Roger. He tries to steal her purse and she drags him home to feed him. Roger is worried throughout the story if she is going to hurt him!
Also there is human v. self in the sense that Roger struggles with the decision of whether to stay in the small apartment or make a run for it. Also, Ms. Washington is also struggling to come to terms with her own past--she too was a troublemaker when she was Roger's age.
In a literary work there is usually a conflict between two opposing forces. In general, there are four types of conflict: person vs. person, person vs. society, person vs. self and person vs. nature. In Langston Hughes short story "Thank You M'am" three of these types are included. First, there is a person vs. person conflict between young Roger and Mrs. Louella Bates Washington Jones when the boy attempts to steal the lady's purse, but is thwarted when she kicks him in the seat of his pants and grabs him. She then scolds him and says she will wash his face. Rather than turn him in to the police, she actually takes him home and gives him dinner. In her apartment Roger also experiences a conflict within himself. On one hand he wants to get away, but on the other he wants her trust him because she has treated him decently:
The woman did not watch the boy to see if he was going to run now, nor did she watch her purse which she left behind her on the daybed. But the boy took care to sit on the far side of the room where he thought she could easily see him out of the corner of her eye, if she wanted to. He did not trust the woman not to trust him. And he did not want to be mistrusted now.
Finally, while never explicitly stated, there is a conflict with society. Roger and Mrs. Jones live in a poor neighborhood where young boys such as Roger are allowed to roam the streets and have little to look forward to at home. Roger readily admits that, even though it is dinner time, there is nobody home at his house. In spite of this poverty, Mrs. Jones retains her dignity and looks to pass on her morality and self-respect to Roger who is obviously neglected by his parents.