A young, unmarried woman is a damsel. If you've ever seen a young girl being chased by a rabid dog, then you've witnessed a damsel in distress!
The word damsel is a shortened version of the French word, mademoiselle, which is what the French call a young woman who is not married — like the word miss in English. Damsel was originally a word reserved for noblewomen — the rich and fabulous. Nowadays even regular unmarried Janes can be referred to as damsel, though you're really most likely to come across the word damsel being rescued by a prince in a fairytale.