Firmware is something in-between "hardware" and "software" - hence the name "firmware".
Hardware, in high-tech devices like phones, is any part of the phone that is physical and cannot be changed unless physically replaced. For example, most Verizon phones are LTE, while most T-Mobile phones are WCDMA. The two systems are incompatible. The Verizon phone contains an LTE chip that is hard-wired to only transmit and receive LTE signals. It could never be changed/upgraded to handle WCDMA signals.
Software, meanwhile, refers to things that can be changed. In a computer, you can of course load new software whenever you want. You can even load a whole new operating system. The software is stored on in memory chips while in use, but as soon as the computer is turned off, whatever was in the memory chips is gone, but a copy is saved on the hard disk.
Phones need to store various non-permanent things, too, such as an operating system software and roaming instructions. But it wouldn't be good if your phone lost its operating system (and thus became useless) every time you turned it off.
Which is where firmware comes in. Firmware uses special memory chips that can hold information even when the phone is turned off and battery-less. It's sort of like hardware, because it acts just like a permanently-wired chip, but it's also like software because it can be changed at any time.
Sometimes operating system updates are available that fix bugs in the way the phone works. Sometimes updates even add new features. These are firmware updates.