In case you are unfamiliar with the local geography:
What people refer to colloquially as "towns" are actually villages and hamlets, which are within actual towns; and, because villages and hamlets are referred to as "towns", then, many times, the error is compounded when actual towns are referred to as "townships".
Also, many colloquially refer to a "downtown business district" in a hamlet as a "village".
NYS Geographic Glossary may be of some help to you in understanding the geography of New York.
You can get a fact sheet for any city, village or hamlet in New York by going here
American FactFinder and input the name of the city, village or hamlet for "city/town" (leave out the ZIP Code) and New York for "State", and then click "GO".
You'll get a lot of demographic, and some economic, statistics for the city, village or hamlet you selected, and if you click on "Reference map" (it's on the right hand side), you'll get a map.
When you do find a house that you like, you can find out in which community (city, village or CDP) that house is actually located, which is oftentimes different from the community named in that house's mailing address, by using the Census Bureau's
online address search function. (CDP, or
Census
Designated
Place, is the Census Bureau equivalent for a hamlet in New York State.)
And, very importantly, among other things, the Census Bureau's
online address search function also indicates in which school district an address is located.