In 1978 Intel released the Intel 8086. It was released under some pressure, as competitors were already pushing out 16-bit design and some 32-bit designs were right around the corner. At the time Intel had no 16-bit processor.
In order to give itself an edge, Intel released the 8086 as compatible with programs written for its earlier 8-bit processors. This was the beginning of the x86 instruction set architecture. It impressed IBM, who decided to use the 8088 (a variant of the 8086) as the processor in the first IBM PC.
This wasn't enough to get Intel out of the woods. In this area of history Intel was still just another player, one among many companies in the new business of making CPUs. Companies like National Semiconductor, Motorola, and even AT&T had their own processor designs which performed as well or better. Early Apples, for example, used a Motorola processor. This pressured Intel to continue making backwards compatible processors including the 32-bit 80186, 80286, and 80386 processors which arrived in the early 1980s. This backward compatibility formed the x86 instruction set architecture.
Through the late 1980s and the early 1990s Intel gradually became the processor of choice. The adoption of the x86 instruction set architecture started a snowball effect which eventually lead to all modern CPUs in personal computers being x86 based, as well as many processor in other devices. Having a set standard for programming CPUs made it easier for programmers to make software and for consumers to make informed choices.