There are a couple easy ways to find out what elementary school zone you live within. You can call the school and ask them. You can call 311 and give them your address and they will tell you everything you need to know. You can go to the DOE website. In the middle of the homepage and choose "find by address". You will be directed to a map that shows your address and your zoned school. Some websites show maps of the zones. These are generally accurate but you need to be careful particularly if you live on a border street, because you may be on the wrong side of the street or at the wrong address for the school that you want.
The school zone is the small (weirdly shaped) catchment area around a school. The district is the larger area that encompasses many school zones. An example of a district is Williamsburg/Greenpoint. Both neighborhoods make up district 14. The district boundaries come into play at the middle school level. Many districts no longer have zoned middle schools. They have district wide choice. This means that if you live within the district you and your child will tour and rank schools in the fall of 5th grade. The schools will also look at many aspects of your child and there will be a computer match. At the high school level there is city wide choice.
At all levels there are also a number of other choices, charter schools, gifted and talented programs, magnet schools and schools that are below capacity and may accept children from outside of their zone or district. I give an overview of many of these choices and the registration procedures for elementary school in my "Intro to Public School: Prek and Kindergarten" talks. I speak about middle school and high school in the spring and summer.