In the early years (1950-1965) IBM made many models of mainframes, many of which were incompatible with each other. It had two main lines of models: one for data processing use, and another one for scientific use. The two lines were largely incompatible. This all changed with the introduction of the System 360 in 1965. The System 360 was a single series of models, largely compatible with each other, for both commercial and scientific use. The System 360 was later succeeded by the the System 370, the System 390 and most recently the zSeries.
The primary operating systems in use today on IBM mainframes include MVS (also called OS/390 or z/OS), VM/CMS (also called z/VM), VSE, TPF, ALCS, and most recently Linux. (A few systems also remain that run MUSIC/SP). Historical operating systems for the System 360 and its successors have included OS/360, MFT, MVT and SVS.
Many of the pre-System 360 mainframes also had operating systems, but they were very rudimentary by today's standards.