In a brave way. It's the adverb of courageous, so it describes how someone did something: she courageously refused, he fought on courageously.

It sits where the adjective sits — formal, weighty, a little grand. You'd find it in a news report, a citation for a medal, a eulogy. In everyday speech people just say bravely, and bravely is by far the more common word.

There's a habit worth pointing out, because you'll meet it constantly and it's easy to misread. Newspapers love "she fought her illness courageously" or "he battled cancer courageously." This is a set phrase, almost a formula, and it's used respectfully about someone who is very ill or who has died. It doesn't really claim anything about how the person behaved on any given day; it's a way of honouring them. Knowing it's a formula stops you from taking it too literally.

Position is flexible: before the verb, after it, or at the start of the sentence for weight. "Courageously, she said no."