When I boot up Linux Mint from a Live CD, I am able to save files to the "File System". But where are these files being saved to? Can't be the disc, since it's a CDR. I don't think it's stored in the RAM, because it can only hold so much data and isn't really intended to be used as a "hard drive". The only other option is the hard drive... but it's certainly not saving to any partition on the hard drive I know about, since none of them are mounted. Then where are my files being saved to??
Believe it or not, it's a ramdisk :) All live distros mount a temporary hard disk in RAM memory. The process is completely user-transparent and is all because of the magic of Linux kernel.
The OS, in fact, first allocates an area of your RAM memory into a virtual device, then mounts it as a regular hard drive in your file system.
Once you reboot, you lose all your data from that ramdrive.
Ramdrive is needed by almost all software running on Live CDs. In fact, almost all programs, in particular desktop managers, are designed in order to write files, even temporary, during their execution.
As an example, there are two ways to run KDE on a Live CD: either modify its code deeply in order to disallow you to change wallpaper etc. (the desktop settings are stored inside ~/.kde) or redeploy it onto a writable file system such as a ramdrive in order to avoid write fails on read-only file systems.
Obviously, you can mount your real HDD or any USB drive into your virtual file system and make all writes to them permanent, but by default no live distro mounts your drives into the root file system, instead they usually mount into specific subdirectories like /mnt, /media, /windows
Hope to have been of help.