Photo by myself, somewhere downtown.
Small convenience stores are universally called "deli's" in New York, regardless whether they have a deli (delicatessen) counter.
They often sell a hodgepodge of stuff, including fresh cut flowers, snacks, batteries, gum, newspapers and cigarettes. Most neighborhood delis are one-offs and don't belong to a chain. There are some Seven-Elevens, but they are rare.
In the photo above, there are a few signs labeled 'dog post'. This is not a common term. I'm assuming this means that dogs are not allowed in the store and there are hooks available to park your dog.
Related posts: On the Pricelessness of NY Delis, A Dollar and a Dream, and All You Can Eat.