To add to this
/system is where the OS lives, as stringer suggested. When you 'install' or 'flash' a ROM you are effectively unzipping the zip file and pushing or copying the files into a given partition. The suggestion to format /system first ensures that there is nothing on that partition from previous ROMs that isn't in the new one that could mess with things or take up unnecessary space.
For example, if you were running CM10, which we all know has a ton of stuff in it feature wise that isn't in say an AOSP based ROM. Then you decide to move over to an AOSP based ROM, you'll effectively leave behind any files in the /system partition that don't exist in the new ROM.
Works the same way on a computer...if you copy 10 files from 1 folder, and 10 from other you have 20 files. If 5 of those are the same from both locations you'd have 15, 5 that got overwritten and 10 more that were unique from the two folders.
Most of the time not wiping /system isn't a huge deal but over time if you don't, you'll chew up space on your /system partition as more and more files get orphaned and never removed.
At least that's how it was explained to me
But yeah, nandroids are stored on the CWM folder on your SD card. So long as you don't manually delete the clockworkmod (not sure where TWRP stores it) folder or relock/unlock the phone you should be good. There's also a blobs folder in there too. That's a differential app backup folder. Don't mess with that one. If you remove it, you're app backups will be rendered useless. The theory behind it is that if you have 4 backups...and 20 of the apps between them are the same, to save space, CWM only backups up one version of the app (not it's data) in the blobs folder that's shared by all the backups. This reduces the speed and space of the backups.
hope that helps!