Originally released as a launch title for the system, the game was a starting point for the the team behind Super Mario Bros. 3 to adapt their work to a new system (and indeed, the game had the subtitle "Super Mario Bros. 4" up until very late in the development process, as evidenced by television commercials promoting the game). The game was originally intended to be released on the NES with a graphics enhancing chip in the cartridge (like Super FX in Star Fox 1 and Super FX 2 in Yoshi's Island), before ultimately moving onto the new system. Other scrapped elements:
The game would have originally taken place in a much smaller world (two islands, which looked to be similar versions of Donut Plains and Chocolate Island), and Lakitu and/or a blue sparrow would appear periodically and fly around Mario (though their purpose is still unknown).
There would have been several areas where the player would have to dismount Yoshi (or have the dinosaur disappear); in the final version Mario only dismounts Yoshi before going through a ghost house, fortress, or castle, and Yoshi returns for the next normal level after that.
At least three of the Koopa Kids were intended to be present on the overworld map (in very different locations that any corresponding level in the final game), and would "pull" Mario into stages (much like the hands in World 8 of Super Mario Bros. 3). Although their access methods were removed, their original positions can be viewed in Lunar Magic.
There were many sprites from Mario Bros. 3 that were intended to carry over into World, including the original Pirahna Plant, Venus Fly Trap and Raccoon Mario (as seen in beta screenshots and the SNES Test Program cartridge).
Yoshi itself went through several changes in production - starting off as a frog, then a western dragon, an ostrich, and finally Yoshi - who incorporates aspects of the first three creatures.
There were at least two other Donkey Kong games planned for the NES - one was a simply musical spinoff, while the other was known as Return of Donkey Kong. While there is little word on this one, common speculation is that the game was ported to the Game Boy. Regardless, Donkey Kong '94 was considered one of the better original titles often overlooked. The game was going to get remade again as Donkey Kong Plus with GBA-GCN connectivity, but was dropped in favor of Mario vs. Donkey Kong.
Super Mario's Wacky Worlds was originally meant to be a CD-i exclusive sequel of sorts to Super Mario World, where Mario would travel through various Earth-based locations. However, it got canceled due to various reasons, and ended up replaced by Hotel Mario, also known as one of the worst games in the series, which ended up in Canon Discontinuity.
In addition to Wacky Worlds, there was another CD-i Mario game titled Mario Takes America, which was going to be an educational game about American culture using live-action Full Motion Video. There was also apparently a Donkey Kong game planned, though nothing is know about it beyond a guy listing it on his Linkedin resume.
Mario Clash, a pseudo remake of Mario Bros.. for the Virtual Boy, was originally conceived as a sequel to the Super Mario Land games for the Game Boy. View a beta video here (skip to 3:12).
Mario and Sonic at the Olympic Games was originally going to feature Silver the Hedgehog, Jet the Hawk, Birdo, Kiki, and Donkey Kong as additional characters, as discovered here. In Mario and Sonic at the Olympic Winter Games, Silver and DK were elevated to playable characters, but Kiki is still MIA. However, you do race against Jet in the Festival, and Birdo is in crowd shots and eventually becomes an opponent in 2012 London.
Super Mario 64 was originally going to be made entirely of more linear levels in the style of the Bowser stages.
Nintendo Power at one point mentioned a sequel to Super Mario 64 for the ill-fated 64DD that would have had Luigi as a playable character, which didn't happen until the DS version of the original game (and the first time he was playable in an original 3D game was the Super Mario Galaxy games).
There were also supposed to be thirty-two worlds rather than the fifteen that were shipped. Players were also supposed to be able to take control of both Mario and Luigi and complete different parts of the castle simultaneously, which obviously means the game was originally a multiplayer game. Yoshi was also supposed to have had a much larger role in the game rather than just hiding on the roof of the castle.
Several years later Super Mario Galaxy and especially its sequel are pretty heavy on linear levels. Super Mario Sunshine and Super Mario Galaxy 2 let you ride Yoshi. The Mario and Luigi co-op still hasn't been seen in 3D. Also, Rosalina was originally going to be portrayed as a relative to Peach.
Galaxy itself, as revealed by Miyamoto in an interview with Famitsu, was going to have more powerups and features that were left out due to time constraints. They were saved for the sequel.
The sequel to Galaxy was supposed to have a more developed story and expand upon Rosalina's history. However, Shigeru Miyamoto, in a rare example of Executive Meddling from him, decided against the idea and wanted the game to focus more on the game play itself as Super Mario Bros. had done. This explains why Rosalina only appears in the ending.
Also, according to a Super Mario Galaxy preview on G4 (circa 2006), it was initially going to be a GameCube game.
Some members of the "The Cutting Room Floor" wiki found some leftovers in the Japanese Super Mario Sunshine version which indicate that you got to the levels via train stations. You buy tickets with Sol Coins. Another interesting thing is that the station list mentions levels that were cut out or even never created at all. These are "Hotel Lacrima" (another hotel!), "Battleship Island", "Erto Rock", "Lighthouse Island" and "Flame Temple".
Going all the way back to the beginning: Shigeru Miyamoto wanted to have the game that would become the original Donkey Kong feature Popeye, Bluto, and Olive Oyl (which Nintendo had publishing rights to in Japan). When King Features balked, Miyamoto reworked the characters to include Donkey Kong, Pauline, and "Jumpman".
Miyamoto got his chance to make a Popeye game shortly after making Donkey Kong. The Popeye game was a modest arcade hit and got ported to several platforms while "Jumpman" went on to be the core character of the Super Mario mega-franchise.
A later Iwata Asks interview gave a few more details about this- apparently Mario could have had a rifle, a beam gun, or a kicking move. There were also plans for Mario to be able to fly on a rocket, and later a cloud (which would later actually happen in the American Super Mario Bros. 2, and Super Mario World respectively.)
Retro Studios wanted to do a Mario football game. As the company was formed mainly for games viewing mature audiences (its founder and many employees worked on Turok), Nintendo instead suggested to make a realistic football sim. That game had some development, but got canned as development of Metroid Prime progressed (and that game itself earns its entry above).
Luigi's Mansion was originally going to be the first Mario game ever to be rated T. It was going to feature a much darker story with much more horrifying monsters and a much creepier mansion. The biggest change was, if the game was not completed by the conclusion of one in-game day, Luigi would not save Mario and possibly become cursed himself.
Luigi's Mansion was also planned to be in stereoscopic 3D using an add-on, but when the production of said add-on turned out to be more expensive than the system itself, it was canned. In hindsight, it makes sense that when Luigi's Mansion 2 was announced, it was on the Nintendo 3DS, which has 3D without the need for an add-on.
Mahito Yokota originally composed up to 28 musical pieces for Super Mario Galaxy that had Latin beats. Koji Kondo said they were no good, so Yokota eventually came up with the sweeping orchestral pieces that made it into the final game.
For Super Mario 3D Land, director Koichi Hayashida suggested fairly early on in development to include the Goomba's Shoe as a power-up. This idea was nixed fairly quickly, though.
There were also suggestions of a cockroach-like enemy that can only be squashed by folding up the 3DS.
Mario Kart 64 was, according to the previews, going to have a Magikoopa in the character roster, but he was later replaced by Donkey Kong.
The game Croc was originally pitched by developer Argonaut to Nintendo as a 3D Yoshi game, though Nintendo turned it down. This upset the developers so much that Croc was only released on rival platforms, such as the Playstation, as revenge.