What two major types called IBM-style PC's does microcomputers fall into?

This is a bit of a confusing question. IBM-Compatible computers are a type of microcomputer themselves. Any x86 based processor, such as AMD, Intel, VIA, and Cyrix, are IBM-Compatible.

Other systems also exist, such as older Macs which used IBM PowerPC processors and Motorola processors. There also exists Sun Systems SPARC processors, and a whole host of others.

IBM-Compatibles are the current dominant consumer system.