World's first restaurant

The very first restaurant in the world was opened in Paris in 1765.  A tavern keeper, Monsieur Boulanger, served a single dish -- sheep’s feet simmered in a white sauce.

Boulanger's business was different from other food businesses, like cafes and inns, because Boulanger's business was centered on food, not alcohol (like taverns) or coffee and tea (like cafes).  Customers came to Boulanger's establishment primarily to eat, and this was a novelty in the late 18th Century, where the population ate their meals at home or, if they were away from home overnight on business, at an inn.

Boulanger claimed that his dish restored one's health, i.e., that it was a restorative.  In French, the word restorative is restaurant.  A local food guild (a union monopoly) sued Boulanger in court for infringing on its monopoly on the sale of cooked foods, but Boulanger won and was allowed to continue.  This victory led to the rapid spread of these new restaurants across France.  See Portable Bistro's Food History website.

U.S. first restaurant

The first restaurant in the United States was Delmonico's Restaurant, which was opened in 1827 by brothers Giovanni and Pietro Delmonico.  More precisely, the brothers opened an ordinary cafe in 1827 and then, in 1830, opened the "restaurant francais" in the building next to their cafe.

Conclusion

Delmonico's Restaurant prospered in New York for almost 100 years.  An understanding of the history of Delmonico's Restaurant requires an understanding of the background and times in which it operated