The Social Security Administration's The History of the Social Security Number gives a timeline for the creation of the SSA and shows how the use of the Social Security number (SSN) as an ID number has expanded over time.
It is useful to search more than one online source for the Social Security Death Index because not all sites report exactly the same information and they display it in different ways.
There will be cases where you find individuals with the same name, and/or your relatives' names may not be what you expect. Indirect evidence can help when sorting out which one of these index records might be your relative. If they were not in touch with the family, there might not be obituaries for them, but if they had siblings, examining the obituaries of all of the siblings might help you narrow down when everyone had lost touch. Death notices may be very brief and not give much identifying information. City directories or telephone books might give clues to their residence before death; city directories and employee newsletters might help fill in the gaps. If a person is well-known in their company, you might find a notice of their retirement in the newspaper or in an employee newsletter.
Write biographical profiles and then work your way forward in time in small steps, working from what you know. It isn't always possible to make a large leap from the last known residence to the records that turn up in the SSDI. Narrowing down the timeline and possible locations will save you having to order several different SS-5s (application for the SSN) simply to find out if the search result is really the person you are looking for.
Edited to add: for an important update about what the public can access, see this post by The Legal Genealogist (Posted on December 30, 2013 by Judy G. Russell):
SSDI access now limited
Here’s the big immediate impact, and it’s from the fact that the FOIA
exemption took effect the minute the budget bill was signed:
genealogists should not — I repeat, we should not — order SS-5 forms
(requests for issuance of a Social Security number) for anyone who has
died in the last three years.
The FOIA exemption means that all requests for “information on the
name, social security account number, date of birth, and date of death
of deceased individuals maintained by the Commissioner of Social
Security” of people who’ve died in the three calendar years before the
request will be denied. So there’s no sense in wasting our time or
money sending in tons of requests for SS-5s on recently-deceased