IBM i is an operating system (previously referred to as i5/OS or OS/400) that runs on IBM's Power Systems hardware (previously called iSeries) and IBM's PureSystems, and it is the primary thing that makes the platform so unique. The name IBM i replaced the name i5/OS as of 2 April 2008, and applies to version 6.1 and above. See OS/400 101 Introduction.
IBM i provides a large number of "Open" resources common on other systems, in addition to a few IBM-only 400 languages and database tools. The i stands for integration of these various technologies, including a very robust variation of the DB2 relational database which is woven into the OS. Another such technology is a Posix compatible environment and shell, the "Q-shell", qsh.
Generally, IBM i has traditionally accessed using a 5250 emulator.
Originally OS/400, meaning Operating System 400, was for the AS/400, meaning Application System 400.
Then IBM came out with variants and re-inventing renaming ... see Names ... such as the AS/36 Advanced System 36, which offered an advanced version of S/36 OS SSP running on the AS/36 one of two flavors: