Etymology
The main German word for “courage” is Mut, which is distantly related to the modern English word “mood” (from Old English “mod”, meaning “excitement”, “frame of mind” or “courage” — from which modern German also gets Gemüt for “mind” or “disposition” and gemütlich for “comfortable”). Its corresponding adjective “courageous” is mutig.
The other (but a bit less common) German word for “courage” or “bravery” is Tapferkeit, related to the English word “dapper”, originally meaning “heavy” or “bold”. Its adjective is tapfer.
Related words are kühn (“bold”), tollkühn (“daring” or “foolhardy”), heldenhaft (“heroic”), as well as furchtlos and unerschrocken (“fearless”).
Examples
Verlieren Sie nicht den Mut. Don’t lose courage.
Ihr fehlt der Mut dazu. She’s lacking the courage (to do it).
Es war eine mutige Entscheidung. It was a courageous/brave decision.
Nur Mut! Have courage!