MacOS (the Macintosh Operating System)--specifically "MacOS Classic"--was a computer operating system created by Apple Computer for use in its personal-use-oriented computer systems which began production and sale on January 24th, 1984. The "classic" MacOS (versions 1 through 9) were cooperative multi-tasking operating systems, and were often known amongst computer enthusiasts for their propensity for regular crashes, which required full reboots of the computer systems in order to recover from.
Mac OS X (MacOS versions starting with 10.0) signify a major change in the base software of the MacOS, as these products are based upon FreeBSD UNIX. This affords the MacOS with an unprecedented level of stability. Moreover, the Mac OS X paves the way for Apple's computer system offerings to take advantage of much more modern hardware (peripherals), and also has allowed Apple to transition the processors used by its computer systems from the Motorola/IBM-produced PowerPC G4 and PowerPC G5 to the more-industry-standard Intel Core-class processors.
Arguably, from a purist user's standpoint, the toughest part of Apple's transition from MacOS "Classic" to Mac OS X on Intel CPU's is the loss of the ability to run "Classic" apps on the Intel-based systems. As a workaround, however, emulation software such as Basilisk II or SheepShaver have emerged to allow greater or lesser degrees of support for such legacy applications.