The operating system developed by Steve Jobs’s NeXT that influenced Mac OS X was NeXTSTEP.
NeXTSTEP was the operating system for NeXT computers and was introduced commercially in 1989. It combined a Unix foundation with a graphical interface and object-oriented development tools. The system was designed not only for everyday computing but also for software creation, helping attract universities, researchers, and developers.
Apple acquired NeXT in 1997, bringing Jobs back to Apple and gaining the technology that became the basis of Apple’s next-generation operating system. NeXTSTEP’s core technologies evolved into OPENSTEP and then into the foundation of Mac OS X, later renamed OS X and macOS.
Rhapsody was an Apple development name used during the transition and is therefore easy to confuse with NeXTSTEP. BeOS was a competing operating system, while Mac OS Classic refers to the earlier Macintosh system family. NeXTSTEP is the specific NeXT operating system most directly associated with the later Mac platform.