ROMMON or Read-Only Memory monitor is sort of like the BIOS on your PC. You should only ever see it briefly during bootup and almost never interact with it (unless something is broken or, very rarely, to add support for new hardware or operating capability).
Software on chassis is the operating system and controls 99% + of the device's operational behavior. That is the IOS (or IOS-XE or IOS-XR or NX-OS or ASA software etc. depending on the platform type).
