It was the first console designed for 3d graphics. The Saturn was designed with 2d graphics in mind, while the N64 was too late to the party.

Well, I can see a bunch of factors that played into Sony's success with the PS1. The biggest one I think, though, was launching at the right time. 3rd party companies were, well... not happy... with the iron fist that Nintendo kept over them- from being the only provider to cartridges, to having to compete with Nintendo's games (which were also cheaper at retail), to the cost of reprints, to the sticking with the cartridge. Sega had eroded consumer confidence outside of Japan, with the okay Sega CD, to the laughable 32X, and the travesty of those few games needing both, only to see the Saturn released 6 MONTHS early with no notice to anyone, including 3rd parties. So what was seen was few games for the CD, fewer for the 32X, and almost none for the Saturn. Why buy a Saturn; no games are coming out for it! Coupled with other facts mentioned (high price {$400 US}, hard to program for, etc), and it wasn't looking good. For some reason, this early launch also angered a few 3rd party companies, who cancelled their projects.

Enter Sony. They were able to discard the bad of each side, and take what was left together. While the N64 was $200 versus $300 on the PS1, games being $40 vs $60-70 meant that you broke even at 5 games (or less). (IIRC, the PS1 was also the only home console to have new games routinely cost sub-$50.) With the cheaper price, the consumer also saw bigger games. Within a year, FF7 moved from the N64 to the PS1, sewing up Japan, and thus, Japanese support from other companies, in a time when Japanese support mattered more to a system's success. The consumer saw the games flow, all for less than either other system on the market.

And the 3rd party companies loved it. It was the easiest system to program for, and the easiest to do reprints for. No more were they bound to Nintendo's sole-providership, or Sega's high licensing fees. Sony's promotion of the 3rd party games also helped in easing their own advertising costs. Why could Sony advertise them? Sony's main developer, Sony Imagesoft, wasn't a huge company, and other 3rd parties didn't see them as much of a threat in relation to Nintendo or Sega. Thus, their games could flourish, and as the system took off in the consumer's eyes, so did the sales of 3rd party games.

This isn't to say that Sony didn't take lumps. The PS1 was the first hardware to be sold at a loss initially. Some early PS1s had issues of scratching disks, making them unplayable. But Sony also did some shrewd things to help sew up their victory. They were known to cut licensing discounts for exclusivity. And they pushed that FMVs could be well integrated into a game, something that the Sega CD did not do. Ultimately, they came at the right time after observing the market as a 3rd party company, and used the Sony name to get their foot in the door. The rest, as they say, is history.

Actually console sales were the reason why PS1 was special. Without console sales, 3rd parties wouldn't have flocked to the PS1 and sony would have been royally screwed as they had the weakest 1st party support at the time out of the 3 consoles available.