The major source for demographic data in the United States is the U.S. Census Bureau. There is an excellent guide for the census by Research Data Services in Lehman Library.
The U.S. Census Bureau is a government agency that produces several surveys; the two major ones are:
1- Decennial Census: What is generally called "the census" (or sometimes "Census of Population & Housing"). Survey of the entire U.S. population, performed once every 10 years (2010, 2000, 1990, etc. back to 1790).
2- ACS (American Community Survey): A more detailed survey, performed more frequently than the Decennial Census (different areas on a rolling basis). If the decennial census data is too outdated or not detailed enough, you may use the ACS instead. See: Subjects included in the ACS
Once you know which survey data you need, you must then choose which source from which to get that data.
The NYC Department for the Aging has a couple of reports on the elderly population: