Readers don't learn much about the American, or "the man," as Hemingway sometimes calls him. He seems young and confident. He speaks Spanish. He knows something about Spanish beer and other drinks, like anis and absinthe, suggesting he has at least a moderate amount of worldly experience. He thinks of himself as rational. His primary goal is to persuade the girl to abort her pregnancy, with the secondary goal of getting their relationship to return to normal, which seems to mean traveling for pleasure. He tries to convince the girl by talking around the issue, minimizing it, and explaining things in ways he thinks makes them seem logical. He does tell the girl he loves her, but only after she asks if he'll love her once she has the abortion. Similarly, after she agrees to have the abortion, he tells her he doesn't want her to have it if she doesn't want to—that they could "go through with it" if she wants to.
The Girl ("Jig")
Readers don't know much more about "the girl" than they do "the American. "The girl," or "Jig," as the man calls her, is pregnant. She's trying to decide what to do about this situation. She's upset, and finds reasons to fight with the man in small ways. She is the more emotional of the two, at least at this moment. Throughout the story, and especially at the start, she seems willing to let the man take the lead and care for her: she asks him what they should drink, what the ad painted on the bamboo curtain says, and so on. Despite being upset, she clearly understands her lover very well. She can follow his oblique references and shifting topics without difficulty.
The waitress brings them drinks, and answers basic questions, like the train's expected arrival time. Her main dramatic function is to force pauses or minor redirections in the conversation between the man and the girl.