ENOS Lives: U R NOT E
Never Underestimate The Power Of PlayStation
Long story short, Nintendo didn't think through a contract with Sony by Hiroshi Yamauchi, then-president of Nintendo. The contract gave Sony all profits for a potential CD-ROM add-on which was being developed by Ken Kutaragi. Yamauchi didn't like the deal, but instead of telling Sony that and drawing up a new contract, he instead went with Phillips to develop an alternative CD-ROM add-on for the SNES... a deal which also imploded (resulting in Phillips' split with Nintendo for their own standalone CD-ROM "multimedia" set-top, the CD-i; a messy legal battle also gave Phillips the rights to some of Nintendo's franchises, resulting in the infamous Zelda and Mario CD-i games), and caused Nintendo to spurn both the 32-bit era and the CD-ROM format. Sony, meanwhile, was reluctant to get into gaming, but that move lost Sony face. So Sony had to get into gaming to reclaim its honor.
Thus the PlayStation as we know it was conceived. Sony redesigned the "PlayStation X" from a fancy CD drive for the SNES into a full-fledged console. Developers were getting excited by 3D gaming, so Kutaragi designed the system with that in mind. He also made sure it was easy to develop for, so programmers could get their 3D system right out the gate. Sony also had, by far, the most developer-friendly license in the industry, with low royalties and no "our console and nobody else's" restrictions. This made the system extremely attractive to developers. Combined with great early sales and the larger profit margin, it attracted many developers, and thus began two generations of PlayStation dominance.
However, in North America, the PlayStation had a rocky start. Sony hired a certain Bernie Stolar as head of Sony Computer Entertainment America, the North American arm responsible for licensing content and developers for the PlayStation. Stolar's Executive Meddling, specifically vetoing many JRPG localization releases for the PlayStation in North America and prioritizing sports titles, almost lead to the PlayStation failing in the market. Sony quickly caught on and booted Stolar and voided his policies. The PlayStation began to thrive in the US once other game genres began entering the market soon after Stolar's policies were removed, and with the release of Final Fantasy VII the rest is history.
Incidentally, only the smaller redesigned models usually display the words "PS 1"/"PS One", given that they were released late in the console's lifespan with its successor, the PlayStation 2, on the horizon, thus there was a need to differentiate it from its impending successor. Though Sony has kept the PS One designation for its downloadable "PS One Classics" line.