It's people who are confusing, not the political divisions.
It's really very simple. In New York, counties are divided into townships,
usually called towns for short. A town(ship) is nothing more than a
division of a county.
Additionally, there are other incorporated areas called villages and cities
which are located wherever there is a concentration of houses.
A town(ship) may contain no villages, one village or more than one village.
A village can sometimes straddle the boundary between two towns.