One could argue that Jim and Della Young are flat characters. Round characters are complex, and they would take many, many sentences to fully describe. Flat characters, however, are relatively simple and can be summed up quickly, in one or two sentences. Jim and Della are flat because they are loving and generous, despite their being so poor, and they put each other's needs ahead of their own. This is really all one needs to know to have a good idea of who they are.
In the story, each one sells the thing that is most precious to them so that they can buy a beautiful gift for the other one: Della sells her hair and Jim his watch. Then, when Jim gives Della the bejeweled hair combs she'd coveted, she has no hair to hold them, and when Della gives Jim a golden watch chain so that he need not look at his watch with embarrassment, he has no watch to which he can attach it. These characters are not complex or difficult to understand: they do not dissemble, hide, or have ulterior motives. They are simple: loving and kind, especially to one another, and according to the narrator, this makes them wise as well.