EEPROM is a non-volatile memory on the Arduino. It's like a HDD in the computer, can hold the data when board is turned off, but has too little memory :(. The EEPROM size of Arduino Duemilanove's Atmega328 is only 1024 byte.
Arduino IDE has simple EEPROM library, which has only 2 functions: "read" and "write". Lets write a demo program which will save to EEPROM anything received from serial port and if the 's' symbol is received it will return all stored symbols.