Your question is so good because it may help others who will find this posting here.
The fastest way I have of searching the Social Security Death Index is to go to Ancestry.com
From their website you can click on the words "Social Security Death Index" that appear below the boxes where you can search for ancestors by name. The SSDI at Ancestry.com is free of charge and you may find other things there free of charge, but most of the really "good" stuff you have to pay for by joining. I have joined in the past but I don't keep up the membership because most of the information I can get elsewhere. You can also go to the Latter-Day Saints Genealogy Site and get the same information there, although some genealogists think the quality of the information is not so good with the LDS site.
To get a Social Security card, a person had to have been working or employable in the 1930's since this program did not exist before then. Many of my Canadian ancestors, some born in the 1860's, came to the US and got SS numbers. The SSDI will tell you in what state the card was first issued, where the person was living when they received their last benefit from Social Security, their birth date and date of death. Sometimes some of that information is omitted. What's really cool is that you can send off to the SS Administration from the information available at ancestry.com and get a copy of the actual SS form filled out when that person applied for a SS #. I haven't written off for one of these, but it could reveal much more interesting information.