Learn Danish: Kvinden køber en rød kjole. - The woman buys a red dress.

The adjective rød describes kjole (dress), which is a common-gender (en) word. Because we are talking about a red dress (singular and indefinite), the adjective stays in its base form: rød. If it described an et word, we would add a -t to make it rødt.

Where is the word for "the" when translating kvinden?

Danish attaches "the" to the end of the noun! The base word is en kvinde (a woman). By adding -n to the end, it becomes kvinden, which means "the woman".

Can køber mean "is buying" as well as "buys"?

Yes! Danish does not have a separate "-ing" tense for ongoing actions like English does. Kvinden køber translates to both "The woman buys" and "The woman is buying", depending on what makes the most sense in English.

If I didn't already know the gender of the word kjole, how does this sentence give it away?

Look at the words right before it! The article en and the base-form adjective rød both tell you that kjole must be a common-gender word. If it were a neuter word, you would see et rødt instead.