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In the following sentence, which I encountered when I was learning French on Duolingo (but my native language is not English):

What do you mean, dead?

In my dictionary, there is no description of such use of dead. The closest one is likely the one equivalent to absolutely.

So what does dead mean in these cases? Does it mean the speaker does not understand what the listener was saying at all?

And also is it considered vulgar to use dead in these cases?

In the construction What do you mean, X?, X is "echoic": a word or phrase (or even a complete sentence) quoted from the previous speaker's utterance. The construction may ask for confirmation or explanation of X, or it may express disbelief or shock.

A: Our proposal is dead.
B: What do you mean, dead? As in the boss rejected it, or we're withdrawing it?

A: Bill's dead.
B: What do you mean, dead? I spoke with him just yesterday!

ADDED:
So X—dead, in your example—means just what it ordinarily means in the context in which the first speaker uttered it.

And of course (as Michael Harvey and David Richerby gently point out) X can be virtually anything:

A: I've just finished the Prolegomena to Any Future Metaphysics.
B: What do you mean, the Prolegomena to Any Future Metaphysics? The assignment was the Critique of Pure Reason!

That would usually be said as an expression of disbelief on being told that someone is dead. It means the same as the everyday adjective dead, because that's what it is. You might consider it as ellipsis of "what do you mean, he's dead?"

Here’s how this question might be used:

Doctor: I’m sorry, but the patient is dead.

Patient’s relative: What do you mean, dead?!

Doctor: I’m very sorry for your loss.

In other words, someone would say this if they could not believe another person was dead even after being told of the fact. It’s a rhetorical question; no answer is really expected. In my scenario the doctor could have said “Well, the patient stopped breathing and I don’t hear a pulse and they don’t respond to stimuli” but that would probably be tactless; the relative probably would not want to hear all about how the doctor knows the patient is dead upon learning of the death.

And no, it is not vulgar to say this, since you asked.

I'm surprised that no other answer yet has mentioned that what we have here is a mention, not a use, of the word "dead." Properly punctuated, the sentence would be:

What do you mean, "dead"?

or to use a slightly older punctuation style,

What do you mean — "dead"?

That is, the speaker doesn't know what the addressee meant by the word "dead," and is asking for clarification.

What do you mean by "dead"?

Omitting the quotation marks is just a colloquial shorthand (like using a short pause — indicated by comma or em-dash — instead of the word "by").

The other answerers are absolutely correct that most likely the speaker is trying to convey that he doesn't believe his ears.

He's dead, Jim.

What do you mean, "dead"? You mean, like, unconscious? In a coma? You can't possibly mean that he's actually dead!

Regarding the use-mention distinction, consider the difference between

VADER: Luke, I am your father.

LUKE [incredulous]: What do you mean, "father"?

and

VADER: Luke, I am your father.

LUKE [immediately deferential]: What do you mean, father?