If there was no Sony Playstation, we can assume Sony and Nintendo didn't terminate their partnership and ended up releasing the SNES CD; and eventually the Nintendo Playstation, the SNES with a built in CD drive. This system would have been released only one or two years after the original SNES launched (around late 1992/1993), and might have taken over as the main SKU. Sony would gain profit on each disc sold (which Yamouchi hated in real life, but not enough to terminate the contract in this timeline). This system would have competed with Sega's own Genesis add-ons, and may have been succesful; causing the release and existence of more CD-based titles than we had in real life.
It's tough to say what would have happened after that though. Nintendo and Sony keeping their partnership would mean that while Sega of Japan and Sega of America were fighting within itself to create a version of the Saturn; Nintendo would have basically little competition for a while. With no need to create a game to compete with the fancy 3D graphics of the PS1, Donkey Kong Country would probably never have been made; and if it did, it probably wouldn't have used the expensive Silicon Graphics pre-rendered sprites.
Just the same, the Virtual Boy would probably not have been made either; without the race to create a 32-bit console out in time for 1995, it would have been a pointless and costly diversion from the real next-gen console Nintendo and Sony would have wanted to do for the second half of the 90's. This next-gen console would obviously have used Sony's CD format (which they would still profit from), and because of that Nintendo would not have lost various 3rd party partners to the competition.
Sega's own infighting would still make the Saturn fall apart, so Nintendo would regain the monopoly they had in the NES generation for the second half of the 90's. The Dreamcast would probably also had suffered a similar fate, since the hype for the next-next-generation system from Nintendo and Sony would be immense (which would also have featured Sony's DVD format). They probably wouldn't get real competition until Microsoft created the Xbox.
However, would there have even been an Xbox? The major impression the PS1 left on the gaming landscape was by expanding console game from the traditional kids audience into the new teen audience. By marketing itself as the console you play when you "outgrow" Nintendo; it not only took Nintendo's aging audience, but it expanded the market considerably into an age group that before didn't care about videogames. This teen audience is the same one that would love "mature" games such as Halo and Project Gotham Racing (adult gamers were mostly cornered into the PC market), and the Xbox was branded to appeal to that audience.
Without the market expanding into that age group, if Microsoft decided to enter the console market; they would have had a peculiarly different system. Since the Sony PS2 would not have been taking away the type of game developers that would make games for the Windows platform, some four engineers in the DirectX team wouldn't even have made the pitch to Bill Gates for a DirectX-based game console! However, if we can imagine that the marketing team at MS saw the potential for expanding the market; it might have been them that would have tried to make a system to the teen audience, and they would have had a similarly vast (but not identical) success to that of the PS1.
Anything after that is free range.