IIRC in the past, there were multiple occassions in which a "button" being pressed in error might have started nuclear war. But I think the guy you are talking about is Stanislav Petrov[1] who is known for preventing nuclear war in 1983[2].
Its not entirely clear that Petrov prevented a nuclear war. His job (which he choose not to do) was to report detected missiles up the chain of command. In theory, the Soviet Union policy was to require multiple-source warnings before retaliating, in which case they still would not have launched anything. In practice, they very well might neglected to follow this procedure and proceed to retaliate.
I remember an interview (will try to find a source) where they told an anecdotal story about Petrov.
Some years after the incident, Petrov told he believed that the system at his station was showing a false alarm and wanted to wait for a confirmation from other stations. He is cited with not wanting to sting a bee-hive.
After being introduced the information present at the other stations (most if not all were seeing the same false-alarm) he told the interviewer that if he would have had this "information" at the time of the incident he'd probably decided otherwise.
[1]: http://en.wikipedia.org/wiki/Stanislav_Petrov [2]: http://en.wikipedia.org/wiki/1983_Soviet_nuclear_false_alarm...