There are two main characters in "Snow," by Ann Beattie, neither of whom is named.
The first, the narrator, is probably a woman, although she does not specify that she is—I assume this only because the narrator's lover is described as a "king of snow" and is therefore probably a man. The narrator is a contemplative person, given to thinking back over the life she has lived. She remembers tiny details of her early life with her lover, particularly the minutiae of the cottage they once went out to one winter when they had first fallen in love. She remembers elements such as the grapes on the wallpaper and the chipmunk which would visit them, and she wonders why it is that she and her lover remember the time differently. Her overriding memory is of the snow; she omits inconvenient details, such as the snowplow, which "seemed to be always there."
The second character, the narrator's lover, we know little about. He seems to be the more practical of the pair, in certain ways. He has clear ideas about "how stories should be told" and tells his partner that any story can be made dramatic if certain parts are omitted. He seems not to be taken in by the fantastical stories the villagers told when they came to visit the couple that winter.
It is also mentioned that many of the locals came to pay visits to the couple during their time at the cottage, but only two of these are singled out: the neighbor, Allen, and his wife. Allen and his wife were evidently somewhat close to the couple, as the narrator goes to pay her respects to the wife when Allen dies several years later—presumably, they all kept in touch.