The Famicom was released in Japan by Nintendo on July 15th, 1983. Compared to other consoles released at the time, including the Atari VCS, Intellivision, and ColecoVision, the Famicom was quite powerful, with higher resolution video, more processing power, RAM, and sprites than the competition. At launch, Nintendo released Donkey Kong, Donkey Kong Jr., and Popeye, which were three arcade successes for Nintendo at the time. While ports of these three arcade games were released on home consoles and computers before, they were significantly different due to the processing power differences between the arcade hardware and the home consoles of the era. The Famicom releases were significantly more accurate to the arcade originals, with more accurate sound, controls, gameplay, and graphics. While other platforms were more approximations of what the original verisons looked and played like, the Famicom ports were more like actual ports.
Above is a comparison between three versions of Donkey Kong: The arcade original (left), the Atari VCS version (middle), and the Famicom version (right). While the Atari 2600 has significantly worse graphics, the Famicom looks nearly identical to the original release. The main concessions to the game are the different screen aspect ratio causing the game to look squashed and the absence of one of the original game's levels.
For a fairly long time after the Famicom's release, most games were similar in their basic functionality to Donkey Kong. Games typically had only a small selection of screens, or occasionally one or two scrolling screens. This was mainly a limitation of the Famicom cartridge size, as Famicom cartridges could only hold up to 40 KB of data (32KB ROM and 8KB for the character map). As a reference to how small 40KB of data is, this downscaled JPEG of my Donkey Kong Famicom cartridge (resized to 1000x600 pixels) takes approximately 119 KB of storage, nearly 3 times as much as the size limit.
These size limitations were not much of an issue for the arcade ports which were popular at the time. However, as time progressed, developers had new ideas and wanted to make original games which pushed the Famicom's hardware. Thus, Shigeru Miyamoto and a team of developers began working on Super Mario Bros., which would be the most technically impressive Famicom game to date.
At a time when Famicom games were expected to only contain single screens or a few small scrolling screens, Super Mario Bros. released with a staggering 32 large, scrolling levels. This was possible thanks to very clever coding by the developers to make the levels fit in such a small footprint. The potential for exploration was unlike any other game released to this point on consoles and only rivaled by Pac-Land in arcades. The levels were well designed, with iconic levels such as 1-1 which helped to teach important game mechanics to the player. By starting at the far left edge of the screen, it taught the player that they would need to move right to progress. The lone Goomba taught players they needed to jump over obstables, or else they would be killed and the level would restart. When the player jumped over the Goomba for the first time, they would likely hit their head on a question block, which would release a shiny coin and make a satisfying sound, which taught players that valuables could be hidden within the blocks. Upon hitting a block further to the right, the player would spawn a mushroom, which upon collecting, would power up the player. The brilliant design of Super Mario Bros.'s levels has been largely covered already, so I will not cover its level design in depth much more. however it's easy to see where Super Mario Bros. succeeds in making fun levels that naturally teach game mechanics to players.
Perhaps something even more important than Super Mario Bros.' incredibly large and well crafted world is the amazing controls the game had relative to all other platforms at the time. Nintendo had previously released platformers such as Mario Bros. and Ice Climber, where the controls were very slippery and jumping didn't quite feel right. Momentum could not be adjusted in midair, which could make landing jumps very difficult. Super Mario Bros. was the first game to perfect platforming movement. The physics of Mario's movement feel just right, with momentum and weight that makes running and jumping much more satisfying and less frustrating than in previous platformers. Mario's hit detection was also very generous, as when touching an enemy, if Mario had any downward momentum at all the game would count it as a stomp and kill the enemy, rather than punishing the player. Compared to other games which may have had difficult to judge hitboxes and frustrating combat, the combat in Super Mario Bros. feels very intuitive while still requiring skill. A key compontent of Super Mario Bros. which was very much ahead of its time was to put effort into reducing player frustration. Every death should be a genuine player mistake, rather than the fault of the game itself.
A large part of why Mario feels so great to control is the effort put into creating excellent jumping physics, which as stated before, many action platforming games before Super Mario Bros. lacked. In most games, a jump would happen whenever the jump button was pressed, and the momentum and jump height would be identical no matter how long the player held the button for, or if the player touched the D-pad or joystick during the jump. In Super Mario Bros., Mario's jump height is based on the player's running speed and the time spent holding the jump button, which gives the player much more control over jump height in an intuitive way. If the player changes their direction using the D-pad, Mario will move accordingly, allowing the player to correct a mistimed jump or to make more precise jumps. These differences are key in helping Mario feel much better to control than other games of the time.
Super Mario Bros. released in 1985 and went on to be a massive hit. It was the Famicom's killer app and became the most sold video game of the time, holding the record for many years afterwards. Despite being usurped by later games with the benefit of being on better hardware and having improved player control, Super Mario Bros. still holds up as having great feeling controls, which is why many still go back to play it to this day. In terms of influence, nearly every game released afterwards has taken something from it. Later Famicom games moved on from being single screen arcade style games and featured massive adventures just like Super Mario Bros., setting a new standard of quality for games. The approach to game design with natural tutorials and more forgiving game mechanics is seen in most highly rated games to this day. Nintendo went on to release many more Super Mario games, including the similarly influental Super Mario 64, which set the standard for every 3D game to come. Overall, the effort put into Super Mario Bros.'s expansive worlds and game design make it (in my opinion) the most influential game ever realeased, shaping the entire industry in its image from that point on.
Also, Super Mario Bros. saved the entire North American video game market from a major crash, but that's a story for another time...