Heh yeah, it's a bit of a contradictory term, but it's the only 'generic' way to describe it that I know of. Most of them are branded with the store's name with a suffix like 'local', 'express' or 'metro'
* Convenience Store
* Dollar store (a little more food options at places like Dollar General/Tree/Family Dollar, usually with a small refrigerator section and eggs and milk and juice)
* "Mini" grocery stores, like Piggly Wiggly Express or Dollar General Market
* Stores with mostly food and limited selection like Aldi
* Supermarkets like Kroger and Publix
* Supercenters like Super Walmart and Super Target with full supermarket selection and lots of other stuff