Although it's strictly speaking an on-rails shooter, the option of 1st and 3rd-person gameplay modes, obstacle-course style levels and three wingmen to support you (or rescue) made the game totally new and innovative for its time.
You play Fox McCloud, the squad leader, and are supported by wingmen Peppy, Slippy and Falco (typically bizarre design by Nintendo there). They must defend their home planet Corneria from the invading forces of the emperor Andross, and take the fight to his homeworld to defeat him.
The polygonal design of three-dimensional models gave the game depth that other shooters didn't have. The ability to fire both boost and retro engines allows you to maneuver around or ahead of obstacles before they can get in your way, and this proves particularly handy in boss battles. You are free to move around the entire screen, rather than merely left to right or up and down, so the sense of actual freedom and control is much greater.
The way the player takes damage is also unique. A shield allows for the craft - called an Arwing in the game - to take damage without immediate destruction, which was the norm for other shooters. One of the major advantages is that the Arwing can be rotated using the shoulder buttons, allowing the spinning that results to literally deflect enemy laser shots. However, the wings could also break off if they clipped the ground, affecting the control of the craft and effectiveness of power-ups.
The way the difficulty is set up adds more replay value too. There are three routes to Venom, the home planet of Andross. Each route has its own unique set of planets, moons, or space stations, so although some are harder than others, there's no "play the same game, only harder" feeling, because you effectively get three different games instead of the usual "Easy, Normal, Hard" set up.
Although we take for granted 3D environments in contemporary gaming, this was virtually unheard of at the time, and so Starfox pioneered the use of 3D graphics. It was and is a brilliant game to play. The boss battles are epic, the level design throws in loads of obstacles and requires quick reactions and tactical movement, there are secret bonus levels, there's choice in how you play, the soundtrack is impressive, there's plenty of replay value and it's very satisfying to complete.
If you can track this down, you should own it, play it, and love it!