Digital images of the New York State Death Index can now be found online at Internet Archive. The images are be made available by a group called "Reclaim the Records." Needless to say, the addition of these indexes online will make the task of searching out New York State death records much easier.
The images are arranged by year. To locate them simply access the Internet Archives' main page, and type "New York State Death Index" into the search box. The search will bring up a page with the individual years now available for viewing.
Here, as an example, is a link to the images for the year 1882.
Prior to the age of the internet, the process of looking up a death record in New York State was tedious and probably beyond the reach of many family researchers, especially those who lived far from New York State. The index was made public on microfiche and was available at just a few large libraries within New York State. Finding a record required a trip to one of the libraries, and involved the time consuming procedure of examining each fiche under a microfiche reader. I've done this, and it was not a fun way to spend the day. Ten or more years ago I started a project of extracting all entries regarding those named BREWER, BROWER, BROUWER, from the microfiche copies. You can see for yourself just how far I got with this. With the death records I was only up to 1897.
The New York State Death Indexes do not cover all localities within New York State. Records of deaths in New York City is noticeably absent. The requirement of recording a death with New York State did not begin until 1880, and in the earlier years compliance with the law was limited. For more background and info on New York State's Vital Records please consult this New York State Archives page. The Buffalo and Erie County Public Library also has published some useful information on searching out vital records in New York State.
Obtaining the actual record of death from New York State involves submitting a form to the State's Department of Health in Albany, and paying a fee. For more information on this also see this page at the New York State Department of Health website, and follow the links of interest to you. Here is an online copy of a mail-in form from 2004. And the same form available directly from the NYS DOH.
Thanks to Reclaim the Records, the first steps in searching out a death record in New York State is now a whole lot easier.