Audacity is the noun: the boldness of a person or an act. In older or very careful English it can mean real courage. In everyday English, especially 'the audacity to…', it often means nerve — doing something cheeky, even rude, as if you had the right. 'The audacity of that email!' is not praise. You can also hear a warmer use: the audacity of a hope, a dream, a young artist who tries anyway. So the word sits on a line between brave and shameless; listeners decide from the rest of the sentence. It is common in newspapers and speeches, less common in very simple talk. Though actually, if you only learn 'courage', you will miss the telling-off meaning, which many native speakers use first.