despair vs dismay

despair and dismay both are nouns.

Word	Noun	Adjective	Verb	Adverb
despair	Yes	No	Yes	No
dismay	Yes	No	Yes	No

As nouns, dismay is a hyponym of despair; that is, dismay is a word with a more specific, narrower meaning than despair:

despair: the feeling that everything is wrong and nothing will turn out well
dismay: the feeling of despair in the face of obstacles

Other hyponyms of despair include hopelessness, resignation, surrender, discouragement, disheartenment, pessimism.

despair (noun)	dismay (noun)
the feeling that everything is wrong and nothing will turn out well	fear resulting from the awareness of danger
a state in which all hope is lost or absent	the feeling of despair in the face of obstacles

despair (verb)	dismay (verb)
abandon hope; give up hope; lose heart	fill with apprehension or alarm; cause to be unpleasantly surprised
	lower someone's spirits; make downhearted