Bodega is word borrowed from Spanish and well established as a general term for a convenience or corner store — in Manhattan and in LA, at the very least.
In Chicago when I was growing up in Chicago we just called them corner stores. Sometimes they were referred to as . mom-and-pop shops because many of them were run by a married couple and their extended family. Kids were sent down to the corner shop a couple times a day, for bread and milk or (I remember clearly) cigarettes. No age restrictions at that time. Some of the stores were very bright and clean, but a lot of them were a little creepy, with dusty, out of date cans of green beans and fruit salad. Which puts me in mind of Lilek’s Gallery of Regrettable Food. The problem is, once you’ve looked at the old advertisements and recipes, it’s impossible to forget those images. People really cooked this stuff. And then ate it.
In Chicago corner stores went out of business as large grocery stores came along, but some have survived. In New York they are still everywhere. They’ve got a history.
Consider the southeast corner of Fifth Avenue and 44th Street, where in 1883 you’d find Tyson’s Market, maybe the first bodega, ever. The story of how Mr. Tyson fought the rich and powerful to hold onto his butcher shop/tavern/corner store (found here at the Daytonian in Manhattan) gives you a good sense of how quickly things changed on Fifth Avenue.
Note the tree in the image above, which was taken in winter. It was a landmark of its own, the Old Willow. You can spot it in the image below (also from the Daytonian in Manhattan). Nothing bucolic about Manhattan in 1884.