Audacity is bold nerve that shocks people. Doing the thing nobody else would dare, or would dare admit to.
The word swings between admiration and outrage depending entirely on whose side you are on, so the audacity of a plan can be a compliment and having the audacity to ask is a complaint. That second construction, with the audacity to, is the commonest use. The adjective audacious keeps mostly the admiring sense, and an audacious escape or an audacious goal are both praise.
Audacity is daring. Latin audeō meant to be bold, which gives audāx, bold, and English has kept the boldness while adding a raised eyebrow to it.
Latin
audāx — bold
Medieval Latin
audācitās
Middle English
audacite
English, today
audacity
Built from
From Middle English audacite, from Medieval Latin audācitās (whence -acity), from Latin audāx (“bold”), from audeō (“to be bold, to dare”).
Etymology from Wiktionary, CC BY-SA