Something that's droopy flops or sags limply. A droopy plant might need water — or you might have watered it too much.
Your dog might have adorably droopy ears, especially if he's a hound or a spaniel. You could also despair over your droopy hairdo, especially if you intended for it to spring up in curls instead of hanging limply down your back. You might also feel droopy after running a race on a hot day. The original meaning of droopy was "sad or gloomy," which most likely comes from the Old Norse drupr, "drooping spirits."