It's not so easy. I'll give you an example maybe it will help a bit.
If someone sees a cat and asks his friend if he sees it too, than it would be "Vidíš tu kočku?".
But when he for example is looking for his lost cat and asks his friend if he sees it somewhere than it would be "Nevidíš (tu) kočku?"
Unfortunatelly it doesn't work this way with all verbs. I know this can be confusing, but I think that with some practice you will understand it.
In Czech questions, both positive and negative verb means the same thing. Vidíš kočku? x nevidíš kočku? The latter can in some situation have more of a meaning "are you totally blind? Don't you see that cat?" but it all depends on an emphasis you use.
In return this structure causes confusion. When you use a negative verb and I do not see a cat and I answer NE, do i negate the negative and thus creating a positive or I do not see a cat? The cat sentence is simple enough that it would likely not be confusing but at times this whole structure leads to a follow up question.
Do not you see a cat? Sounds better (for me) and is correct (for Duolingo).
Jimmy, your sentence is not a correct translation, because your English sentence uses 'the', which is not in the original Czech.
I think you were trying to say: Don't you see a cat. That is a correct translation.
Don't you see the cat? = Nevidíš TU kočku. So according to Duo's Golden rule it is a different sentence, not one of accepted translations.
"Do you see the cat?" should be accepted. I see Czech speakers make this direct translation mistake in English all the time. For example, once my boss asked me "don't you have time today (to go additional work)?" this translates directly from "nemáte čas?", but sounds super harsh in English.
FWIW, I (unthinkingly) typed "don't you see THE cat" -- and, luckily, it was accepted! But, as others have said, in the absence of "tu," the English translation really is better with "a" than with "the."
Both are possible here, but that is not always the case.
If you have a cat at home and you are looking for it, then "Nevidíš kočku?" means "Don't you see the cat?". It is a question if our cat is somewhere near. You use "kočku" instead of using her name.
You can also ask when you just saw a random cat and you are asking if I can see it as well. I am not completely sure if that is a cat or the cat in English. With "tu kočku" it is clearly the cat.
In a completely generic situation it would mean "Don't you see a cat?" with the meaning of "some cat". Just a question if you happen to see a cat. Why? Who knows.