People often wonder if a building permit is required by your town. For the Village of Ilion NY, here's some info.
Every town and village has rules governing the building of structures and how the land is used. This is good because it sets some standards and helps neighbors get along.
The Village of Ilion website has their building permit form which states "a permit is not required for a shed under 140 sq ft" (stated on pages 7 and 8). So, for example an 10x12 shed is 120 square feet. Since this is under 140 square feet a permit would not be required. (However, this could have changed since the time of this posting, so you should still check with the Village of Ilion.)
There are similar rules for other towns, but to be sure, check with your own local city or town office.
What should you look for in a Shed?