Temerity, audacity, hardihood, effrontery, nerve, cheek, gall are comparable when they mean conspicuous or flagrant boldness (as in speech, behavior, or action).
Temerity usually implies contempt of danger and consequent rashness; often it suggests, especially when a proposal or project is under discussion, a failure to estimate one’s chances of success.
Audacity implies either a bold and open disregard of the restraints imposed by prudence, convention, decorum, or authority or undue presumption in making advances.
Hardihood stresses firmness of purpose and often additionally implies considered defiance (as of conventions or decorum). It may be used without depreciative intent, but it is frequently employed as a term of contempt almost equivalent to insolence or impudence .
Effrontery is definitely derogatory; it is used in place of any of the three preceding words when one wishes to impute flagrant disregard of the laws of courtesy, propriety, or fair dealing or an arrogant assumption of a privilege.
Nerve , cheek , and gall are close to effrontery , nerve , however, often carrying a strong suggestion of hardihood, cheek of impudent self-assurance, and gall of outrageous insolence.