Usually, on a live CD or Live USB key, all modifications are discarded when you reboot.
The persistence allows you to keep your preferences and data even after reboot.
The data are stored in a special file called casper-rw (for Ubuntu) and overlay-USBLABEL-UUID (for Fedora). This is a completely transparent process for the user.
Usually, 300 MB are enough to install some software but you can use more.