Super Mario Bros. 3 (NES, 1988)

The first Super Mario Bros. game for the NES made leaps and bounds for video gaming as a whole, but Super Mario Bros. 3 (1988) achieved liftoff and soared.  Quite literally, in fact.  This game takes everything that made the original great and builds upon it with grace and mastery.  Aside from the obvious graphical improvement, the game features tighter controls, more power-ups, an improved multiplayer mode, and an all-around better design.  In every imaginable way, Super Mario Bros. 3 improves upon its predecessors.
In this game, Bowser, the King of the Koopas, has returned with seven (mysteriously spawned) children, each given an airship to invade the seven kingdoms and steal the scepters of the seven kings, also turning them into animals.  It's up to Mario (and his younger brother Luigi) to retrieve the scepters and save the land once again.  After doing so, the Mario Bros. discover that Bowser has kidnapped the Princess Toadstool behind their backs, and they must go to his lair to save her.  As with most NES games at the time, the story only serves to tie the game together, and the real focus is on the gameplay.
Oh boy, does this game offer fun in abundance.  Mario controls smoother than ever, aided by his many new abilities.  In addition to the returning fire flower (which allows Mario to hurl fire balls at foes), there is the leaf, which gives Mario the raccoon suit, which allows him to float, whip his tail at enemies, and after running for a short stretch, fly for a given time.  Similar is the elusive Tanooki suit which also adds the ability to transform into an invincible statue for a short time.  This is based on a piece of Japanese folklore that is lost on American audiences.  Japanese myths tell of Tanookis, raccoon-dogs who have the powers of flight and transformation.
Super Mario Bros. 3 ditches the linear level progression of the first two games in favor of a map screen, where each of the eight levels in the game have an overworld to navigate, and select levels you want to play in essentially any order you want.  Some stages can be skipped entirely.  Hammer Bros. patrol the map, and running into one of them initiates a short battle, where victory rewards you with items.  Mario can now hoard items to use on the map screen at any time the player wishes.  Item houses on the map can also be visited to acquire these.  Each map also conceals hidden paths and secrets.
The multiplayer of this game is also something to be noted for its depth of execution.  In the first Super Mario Bros., players alternated turns playing after one of the players lost a life, which could be forever if you were playing with someone who was highly skilled.  Now turns alternate after each player finishes a level.  The game fosters an odd balance of competition and cooperation between the players.  Mario and Luigi must work together to finish the levels and get through each world successfully, but will naturally compete for the best items and stages.  If one player goes to the other player's space on the map, they can initiate a battle game based on the original Mario Bros. arcade game.  The winner will be able to take some of their (former) friend's goodies.
What the Mario games have in common after this title is a feeling of flight that most other games cannot achieve.  Mario's controls are a kind of perfect that is difficult to describe in words.  You just know that when you are playing a Mario game, it feels right.  Mario would continue to fly higher and higher with each successive game, and he would take on many different genres beside platforming as well.  The Mario franchise of games is one of the most profitable and influential in all of gaming, fittingly because it is one of the best.