The main conflict in “Thank You M’am” is character vs. character, but another important conflict is character vs. society, and there are internal conflicts.
The obvious character vs. character conflict is between Roger and Mrs. Jones. He tries to steal her purse, and she makes him come home with her.
She still held him. But she bent down enough to permit him to stoop and pick up her purse. Then she said, "Now ain’t you ashamed of yourself?" (p. 1)
The conflict is resolved because the two characters come to an understanding and appreciation for each other.
The second conflict is character vs. society, because both characters have had a rough go of it. Mrs. Jones and Roger both wanted things they could not have, because society creates a materialistic need.
"I have done things, too, which I would not tell you, son—neither tell God, if he didn’t already know."
Mrs. Jones realizes the draw of the shoes—and she gives Roger her hard-earned money so that he can have what he wants.
Finally, there are internal conflicts. Mrs. Jones has to decide what to do about Roger after he attempts robbery, and she takes the unusual step of bringing him home with her. Roger also has to decide whether to run away or not, and he decides not to.