The firm that I will discuss in this answer is Safeway. It is a large grocery store chain. The market structure that best fits Safeway’s characteristics is oligopoly.
Clearly, Safeway is not in a monopoly. It has competitors such as (in various parts of the country) Albertsons, Piggly Wiggly, and WalMart. This means that it cannot possibly be a monopoly. On the other hand, it is clearly not in perfect competition. In perfect competition, there are a very large number of competitors selling a homogeneous product and competing only on price. This is not the case with Safeway as it has relatively few competitors and also tries to compete on things like atmosphere and amenities.
This leaves oligopoly and monopolistic competition as possible market structures. Of these, oligopoly is the better choice. Markets in monopolistic competition generally have a large number of sellers. The grocery industry is not like that. It might once have been like this when there were many little “mom and pop” grocery stores in every town. Now, however, the grocery business is dominated by a few major chains such as Safeway (which also owns Vons), Albertsons, and Kroger. Because of the small number of competitors in this market, it is best categorized as an oligopoly.