Americans often seem to have two main sorts of places, big urban conurbations and "small towns", which have a connotation of rural life and are often really small (the cultural background to this is uncertain for me, a lot of American cultural artefacts seem to be very nostalgic about small towns). Please, can some American correct me on this: I'm just going by TV and books...
Meanwhile Brits have a few extra terms. "Village" (or occasionally "hamlet") is the usual term for a really small place. Small in population terms, sometimes quite straggling spatially. A village is unlikely to have much tourist trade unless it's really basing itself on that, and that sort of thing involves a lot of thinking about the road network, how the tourists get there and from where. That's the usual pretty, kempt village with a village green and planning permission to keep eyesores out the Americans are often thinking of, but that sort of village needs plenty of well-off people to keep it that way, it doesn't just happen.
Country villages may well have started out based on agriculture and may nowadays be slowly dying as the younger people leave. Villages typically have few or no specialised shops, just a corner shop/newsagent desperately hanging on while people go to supermarkets nowadays, and definitely a pub. What makes a village nowadays is probably rural location (at least some fields between it and the nearest town) and the lack of specialised shops.
Suburbs are often erstwhile villages that got swallowed up by the nearest town: as a rule of thumb, if a place is, say, four miles from a city, you won't see any fields or anything separating it from the town centre. This sort of place probably was a village several decades ago.
One dead giveaway is Americans talking about a village when it isn't. One story threw me right out talking about Godalming as a "pretty little village with whitewashed stones". Er, no. Looking it up on Wikipedia, it's an attractive little market town that probably sort of counts as a suburb (4 miles from Guildford, the county town of Surrey). The Home Counties (i. e. the ones surrounding London) have very high property prices and are thought of as desirable places to live/commute from. A book-blog recommending E F Benson's excellent Mapp & Lucia novels described them as set in the charming little village of Tilling. Nope. Tilling (based on Rye where the author lived) was an attractive market town. A town with enough history and enough prosperous people to support a market (in the sense of open-air stalls). He wrote about people who were distinctly well-off, not to say idle, and could afford to keep in touch with London fashions and theatre. Certainly at the time when Benson was writing, village life would have been a trade-off: you got the fresh air, but lacked civilised amenities!
Towns may be of various sizes, but they're very likely to be bigger than villages, and have more ambitious shops. If it has a chemist's (="drugstore"), or a shop from one of the national chains ("Boots", "Smith's", etc), it's more likely to be a town. If it has a department store like Debenhams or John Lewis's it's seriously big. Bigger towns, often county towns, may be bigger than small cities. Cities get their status when the Queen says so. Earlier on, city status was often down to the possession of a cathedral.
See comments below: I wasn't entirely accurate in the definition of villages, although they are roughly "usually smaller than towns".
The best advice I can give Americans (and, of course, other non-Brits) is to check Wikipedia etc for not only direct status ("village", "town" with subtypes, "city") but a rough idea of the social and economic class. Shading richer goes toward prettiness and being able to afford nice shops/schools; shading poorer will get dying rural villages, the occasional eyesore that never gets fixed, and city/town/suburban shop-fronts remaining uncleared since they went out of business suddenly. Also, even the nicer big towns and cities have horrible areas.