In the early days, "PC" was supposed to be an abbreviation for a "Personal Computer", also known as a "Microcomputer". Lots of (mostly-now-vanished) companies sold different kinds of PCs, such as the MITS Altair, Tandy, Apple and so on. Then IBM brought out its model 5150, more commonly-known as the "IBM Personal Computer" or "IBM PC" (yes Virginia, all IBM machines had four-digit product codes). This family of machines, and compatible ones introduced by other vendors, came to dominate the market so completely that, for many people, the term "PC" became synonymous with the IBM kind. And the compatible ones made by other vendors were, of course, described as "IBM-Compatible" PCs.
But IBM doesn't make PCs any more. But even when it did, it wasn't anything produced by IBM that determined what was "compatible" and what wasn't--it was products from Microsoft.
In about 1985, the standard for determining whether a PC was "IBM-compatible" was whether it could run Microsoft Flight Simulator. In the days before Windows was popular, when graphically-based programs running on these machines had to do their own direct hardware access in order to perform fast animations, MFS was considered to be a good stress-test for true Compatibility.
These days, the compatibility standard is even clearer: it's the ability to run Microsoft Windows and the applications written for it. The current version of Windows, XP, derives from the Windows 2000/NT line, which early in its history was designed to work cross-platform on architectures other than x86, including Alpha and PowerPC. However, those ports were soon abandoned as unprofitable, so these days "Windows" very clearly means "x86, together with a BIOS interface and a whole lot of hardware quirks that date back at least to the IBM model 5170 (the PC AT)". There are a great many instances in the Linux kernel, in particular, where the developers had to work around bugs in chipsets and other hardware, where the only testing the vendor did before shipping was to make sure Microsoft Windows would work.