This is the 14288th most frequent Russian word.
отваге
"Отваге" means "courage" or "bravery" in English.
In this sentence, 'отваге' is in the dative case and forms part of the construction expressing gratitude (благодарность за что), emphasizing the quality being acknowledged.
Мы благодарны за отваге, проявленной спасателем.
We are grateful for the courage shown by the rescuer.
Here, 'отваге' is in the prepositional case as part of the phrase «помнить о чём», referring to a memory of or contemplation about something specific.
Они вечно будут помнить о его отваге.
They will always remember his courage.
The word 'отваге' here is used in the dative case as part of the construction «восхищаться чем», demonstrating admiration or inspiration from something.