despair vs pessimism

despair and pessimism both are nouns.

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

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

despair: the feeling that everything is wrong and nothing will turn out well
pessimism: the feeling that things will turn out badly

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

despair (noun)	pessimism (noun)
the feeling that everything is wrong and nothing will turn out well	a general disposition to look on the dark side and to expect the worst in all things
a state in which all hope is lost or absent	the feeling that things will turn out badly

despair (verb)	pessimism (verb)
abandon hope; give up hope; lose heart