the use of the Arduino's EEPROM
In short:
Once you shutdown an Arduino, every variables on it, which have been calculated by the sketch, are lost. That is because they are stored in RAM, which is not able to store things without power. That is not the problem of EEPROM: values stored there will be available on reboot, and won't need to be calculated again.

This can prove useful is your device is allowing a user to store presets: so a configuration of the program. If this configuration, which can be for instance variables affecting the program behavior, is stored in RAM, it will be lost on next reboot. On the other hand, a value stored in EEPROM during the first execution (entered by user for instance) will still be available on next reboot, if you took the time to write it (on the first run) and to read it (on the second run). Just like saving and then recalling a preset.

Beware that EEPROM does not like to be written too many times and wear out, so you should keep that to a minimum (values that really need to be restored on reboot, and cannot be calculated at this time).