a woman - en kvinde, the article is seperate to the noun the woman - kvinden, the article is included in the noun
There's no trick to learning this other than just remembering it, and you will need to use this a lot, so it's probably something you'll want to get the hang of pretty early on
If you scroll down on this page: http://ielanguages.com/danish1.html No.5 give a basic explanation to the use of articles in the Danish language, though you may want to find a more in depth introduction to Danish Grammar before you go any further. It'll save you a lot of headaches later on down the road.
I thought the question itself was asking for "a" woman. Upon entering that i was told it was wrong as the translation was "She is 'the' Woman". Hun er kvinden.
There are two things wrong with your English translation. "Kvinden" means "the woman" (lady can be used as a synonym here which is why it appears as a correct answer, I don't know why it thought this was the closest to your translation), "a woman" would be "en kvinde". Also "kvinden" is singular, but "women" is plural, the singular form in English is "woman", "women" would be "kvinder" and "the women" would be "kvinderne".