What is Cisco IOS and what does it do?

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Accepted Answer

Internetwork Operating System — the operating system that runs on Cisco network devices and is driven through a command-line interface (CLI).
Just as a PC runs Windows or Linux, a Cisco device runs IOS (Internetwork Operating System). It is the software that boots the device, manages its hardware and memory, and — most importantly — provides the command-line interface (CLI) you use to configure and monitor it. The same IOS family runs across Cisco switches, routers, wireless access points, and firewalls, so the commands and modes you learn transfer from one device to the next. When you set a hostname, a password, or an IP address on a switch, you are issuing IOS commands.
(Aside: "IOS" is a registered Cisco trademark; Apple uses the lowercase "iOS" name under license.)
Go deeper:

Cisco IOS (Wikipedia) — history, architecture and the CLI-mode model shared across Cisco switches, routers and firewalls.