Look up Monty Python's 'Dead Parrot Sketch' for a list of euphemisms for "dead". :) Pinko has given a great list of euphemisms, but the warning is that these are NOT all interchangeable. The possible list is endless, so I'll just explain what you ask about:
"Died" is the past tense of "to die", so this is direct.
"Passed away" ("or passed on") is a polite way of saying someone has died. As though they decided to go somewhere else. Much less direct than saying "died".
If you hear someone's relative has died, you would say "I'm sorry to hear that he/she has passed away". It sounds much more courteous.
"Gone" has the same polite meaning as "passed away", but you would use this to report the person has just died. That is, they were alive last time you heard news about them, but you knew they were about to die (ie. they were "going").