The Legend of Zelda (NES, 1986)

Shigeru Miyamoto is one of the most recognized names in video games, and for good reason.  He has created many games that have become mainstays in craft and popular culture.  His titles shape the way other video games are made.  From Donkey Kong to Pikmin, he has repeatedly redefined the way we play games.  His most important contribution to the medium, I believe, was 1986's The Legend of Zelda, which has had an influence on every modern game, whether or not they realize it.  It was inspired by memories of his childhood, where he would run around the woods near his home and pretend to fight monsters and go on adventures.
At the time, most games were designed to be played in one sitting.  This was because most games were played in the arcade, and if the game was too long, no one would be able to finish it.  Most arcade games didn't have endings, you just tried to get the high score.  With home console games though, there was more room to experiment with games that required extended playing.  The Legend of Zelda told the story of Link, a young boy who is destined to defeat the evil Gannon and save the Princess Zelda.  The three of them make up the trinity of the Triforce, each of them carrying a part of it inside them.  However, Link must find the eight pieces of his in order to truly gain its power.  To find them, you must locate and fight through eight dungeons, then the final dungeon, where the legendary showdown with Gannon occurs.
This was one of the first games to have a battery that allowed you to save your in-game progress and pick up where you left off later.  This was entirely necessary since Zelda is a LONG game.  Part of the length of the game comes from the fact that you are not told specifically where to go.  There are a few characters who give you hints (some at a price) but much of your progress must be achieved on your own merit.  This is another influential part of Zelda, its non-linear story progression.  You can essentially do the dungeons out of order, but some dungeons can only be accessed using items found in previous dungeons.  For example, in dungeon 3, you find the raft which is needed to float across the lake to get to the entrance of dungeon 4.  I usually do dungeon 3 before dungeon 2, since it is easier to find.
Games from the early NES days are usually found to be archaic in nature, usually because of old-fashioned design (see my Dragon Warrior review).  The Legend of Zelda is one of those few games that does not feel like it has aged a day.  The graphics have a timeless aesthetic, the music is still as fresh and memorable as ever, and the play of the game is still perfect.  It stands the test of time, and can still hold a candle to today's games.  The golden NES cart is a game that is sincerely worth its weight in gold.