belief vs idea

belief and idea both are nouns.

Word	Noun	Adjective	Verb	Adverb
belief	Yes	No	No	No
idea	Yes	No	No	No

As nouns, idea is a hypernym of belief; that is, idea is a word with a broader meaning than belief:

Other hypernyms of belief include thought.

belief (noun)	idea (noun)
any cognitive content held as true	the content of cognition; the main thing you are thinking about
a vague idea in which some confidence is placed	your intention; what you intend to do
	an approximate calculation of quantity or degree or worth
	a personal view
	(music) melodic subject of a musical composition