In the US, a deli is a store selling cold cuts, cheeses, and a variety of salads, as well as a selection of unusual or foreign prepared foods. I imagine, if you are looking at a menu, and something is served deli style in the US, it probably means that it is the kind of sandwich you'd find in a deli.

A delicatessen, or in short, a “deli,” is a store where people can go to buy ready-to-eat items such as cold cut meats, sliced cheeses, sandwiches, breads, salads and so much more. If you're from the Mid-Atlantic or New England areas, you probably have a favorite deli — or a list of at least a few favorites!

Quality & Freshness Paulie James, owner of the popular Los Angeles deli Uncle Paulie's, says that a combination of "high-quality ingredients and top-notch bread that complement each other well" sets good deli sandwiches apart from the rest. "When you use fresh, high-quality ingredients, you can taste the difference.