On October 17, 1985 (by one blog's reckoning [edwardbosworth.com]; an archived Infoworld article [google.com] mentions October 1985, but not the day of month), Intel released the 80386 CPU [wikipedia.org], a 32-bit chip that supported 4 GB address spaces, memory protection, and virtual memory, while retaining the ability to run 16-bit MS-DOS programs in one of two compatibility modes ("real mode" for the original 8086/8088 instruction set; a 16-bit protected mode for the 80286 introduced with the IBM PC-AT). The 32-bit protected mode also supported multiple "virtual 8086" modes, or emulations of 16-bit real mode that each ran as a hardware-protected domain.
The 80386 was certainly not the world's first 32-bit CPU; DEC began selling 32-bit VAX [wikipedia.org] computers in 1977, and many of its competitors in the minicomputer and workstation markets followed suit with their own proprietary 32-bit processors. Even in the personal computer space, the Motorola 68000 that powered Apple's Macintosh (famously introduced during the Super Bowl in January, 1984) was arguably a 32-bit CPU, although it had a 16-bit external data bus.
Intel's new chip should've taken the PC industry by storm, and it eventually did. But MS-DOS, the dominant PC operating system, was based on the original 8086 instruction set which could only make effective use of 1 MB of RAM. A series of hacks developed by engineers throughout the PC industry, including bank-switched "expanded memory" and "extended memory" (switching processor modes on the fly), raised this limit a bit, but these workarounds were far from transparent to application programmers. IBM's response was the OS/2 operating system [arstechnica.com], co-developed with Microsoft, that could run in either 16-bit (80286) or 32-bit (80386) protected mode, thus taking advantage of additional installed RAM, in addition to the obvious benefits of memory protection. The trouble was that OS/2 couldn't run the complex MS-DOS apps that business and home users were depending on; Bill Gates figured this out, but the IBM executives didn't. And when IBM released its PS/2 line of PCs in 1987, the high-end models equipped with 80386 chips were ridiculously overpriced.
Meanwhile Compaq, one of the vendors who had figured out how to clone and sell an IBM PC for much less than IBM's asking price, hit upon the idea of using the 80386 as a way of distinguishing their product from the competition. Then Microsoft released Windows 3.0 in 1990, with three installation modes corresponding to the different Intel chip architectures (real mode, 80286 mode, and 80386 mode), but it was the 80386 mode, and chip, that moved the market and caused a flurry of hardware and software upgrades.
Intel eventually licensed production rights to the 80386 to several competitors, including AMD, Cyrix, and IBM; Intel finally ended its own production run in 2007 (see Wikipedia article).