ROM - Read Only Memory. Real ROMs are programmed at the chip factory and cannot be changed.
PROM - Programmable Read Only Memory. These can be programmed outside the chip factory but only once.
EPROM - Eraseable Programmable Read Only Memory. These are ROM chips that can be erased by shining UV light into a clear window on the top of the chip exposing the innards. A dedicated EPROM eraser device is typically used that uses a timer to expose the chip to UV light for the proper amount of time. EPROMs are programmed by a special "burning" device that typically connects to a computer via a USB or printer port (older devices). The new data is "burned" into the chip.
EEPROM - Electronically Eraseable Programmable Read Only Memory. EEPROMs can be erased and reprogrammed from within a device (no eraser or programmer required). The chips are typically implemented using Flash memory and are used in all kinds of portable electronic devices such as cell phones. Memory cards typically contain EEPROMs.