October 1984 – RFC 920 establishes 7 generic “top level domains” (gTLDs, including .com, .net, .org and .gov) to provide domain space for corporations, non-profits, schools, networks, US government offices and the US military
To the best of my knowledge the 1st domain is not a .com but as .net as Steve said elsewhere:
NORDU.NET Creation Date: 01-jan-1985
Some ccTLDs were delegated shortly after, like .se in 1986 http://twobirds.com/English/publications/articles/Sweden_regulation_national_top_level_domain.cfm
However, NORDU.NET is probably the oldest of all domain names.
I believe DNS was devised and 1983 and the first DNS servers were put in place around 1984.
Most people associate domain names with the Web, which emerged around 1993 but corporations/universities were already using other stuff like e-mail or FTP well before