Super Mario 64 (Nintendo 64, 1996)

By 1995, the Sony Playstation, a 32-bit console, had been released, and started revolutionizing gaming with it's polygonal 3D graphics and gameplay.  Games like Crash Bandicoot and Tomb Raider made some of the first steps in making games in the third dimension, but in retrospect, did not make much actual progress.  Crash Bandicoot (while it is a fun early Playstation title) was basically a 2D platformer in disguise, and Tomb Raider features controls so wretched as to be almost unplayable today.  Then the Nintendo 64 arrived on the market, bundled with Super Mario 64 (1996) and for the first time, a 3D platforming game felt right.
Mario's first 3D adventure has him on a routine visit to Princess Peach's castle (she has written him a letter promising cake) when he discovers that Bowser has already infiltrated the castle and hidden away with her somewhere within.  Bowser has also stolen the castle's 150 magical stars, and placed them inside of the paintings spread across the castle.  When Mario jumps inside of one a painting, he is transported to another land, where he must use his new skills to reclaim them.  Locked doors can only be opened if Mario has enough stars for them (a door may have a "3" on it, meaning you need three stars to open it.)
The first thing I suppose I should talk about with Super Mario 64 is the level design.  Each level (including the castle) are like huge playscapes, filled with secrets, all fostering exploration.  Peach's castle is one of the best and most important overworlds in gaming.  An overworld takes the place of a level select screen, and can basically be a level unto itself.  The main parts of the castle are all iconic, but its secrets are what we remember most.  Consider the room that is seemingly empty, save for three stained glass windows.  Jump into the one on the right and you will discover a secret level, a huge slide that rewards a star at the bottom.
Each level contains six stars (and another for collecting one hundred coins) that you need not do in any order.  Some stars can be completed on your first visit to a level, and some will require you to unlock new abilities before you can get them.  Mario has three power-ups he can earn and use throughout the game, in the form of magical caps.  The wing cap allows Mario to jump higher and fly, one of the greatest joys in the game, and all of gaming.  The metal cap turns Mario into solid steel, making him invincible, and able to walk underwater, but he is slower and less agile.  The vanish cap makes Mario transparent and able to faze through solid objects.  Sometimes, these caps must be used in junction to get stars, such as the one near Bowser's submarine that is in an underwater cage.
The levels would all be worthless if not for Mario's controls.  Just like how Mario perfected the art of 2D platforming, so does he for 3D platforming.  The N64 controller uses an analog stick, giving Mario precise directional movement.  You may run in 360 degrees, steer Mario in mid-air, and tilt the stick at different angles to control his speed.  Some enemies may only be approached (like the sleeping piranha plants) by tip-toeing up to them.  Beside jumping, Mario has been given a multitude of new actions to perform.  He can punch, kick, do a ground-pound, jump off of the walls, climb, and much more.  Every move imaginable is at your disposal, giving you full reign on this new, 3D world.
The whole game is filled with iconic and memorable moments.  The first time you jump into a painting, and hear the triumphant music of Bob-omb Battlefield.  Exploring the eerie Boo's Mansion, Mario sees a red coin (eight hidden in each level for a star) sitting behind a grand piano.  When he goes to collect it, the piano springs to life, gnashing teeth inside it, making a fearsome noise of sour notes.  Having to creep up next to that creepy eel  near the sunken boat to lure it out, and then trying not to drown inside of the ship while you solve a puzzle.  The first time you faced off against Bowser (there are three encounters), having to grab him by the tail and swing him at one of the mines in the room.
Super Mario 64 is the first video game I ever encountered.  I went to a friend's house, and he had a Nintendo 64.  I had never heard of video games before.  My life was forever changed when I experienced the joy of Mario's acrobatics and marvelous adventure.  When I went home later that day, I tried to recreate the experience by hopping around on the furniture.  It didn't work.  Later that year, on my birthday, my parents got me a Nintendo 64 with the Mario game, and a lifetime passion was born.