"What does it mean?" vs. "What does that mean?" — what's the difference?

Solution 1:

Both are acceptable. That is more specific. You would probably use it for a very particular thing whose meaning you wanted to understand. For example:

1. You are looking at a sign written in a foreign language and you turn to a native speaker and ask "What does it mean?"
   

2. You are looking at several signs. You understand most of them but there's one you don't understand. You point and say "What does that mean?"

It is more general. What does it mean? (life, the universe, everything)

Like your friend says, it depends on context, but actually they are fairly interchangeable. Everyone would understand you.

Solution 2:

Generally speaking, the word that is simply more pointed than the word it, but not necessarily. Compare the following pairs:

Me: "I'm totally discombobulated."

You: "What does that mean?"

Compare the following:

Me: "I'd use the word discombobulated, if I were you."

You: "What does it mean?"

Pretty similar, yes? Look a little more closely, however. The first pair indicates that you want to know what aspect of the word discombobulated applies to me, and not necessarily a definition of the word. The second pair indicates you want me to give you a definition of the word discombobulated. There's a difference between the two, but it is subtle, to be sure.

On the other hand, the two words (it and that) are sometimes not interchangeable, especially when the word that takes the place of several (or more) words.

Me: "You look as if you just bit into a lemon!"

You: "What is that supposed to mean," you say defensively.

Here, the word that takes the place of "you just bit into a lemon," and you might rightly be a little offended because I infer you are calling me, indirectly, a sourpuss!

Compare the following:

Me: "You look as if you just bit into a lemon!"

You: "What is it supposed to mean?"

Here, the word it is not used appropriately, since the referent for the pronoun, it, is not clear.

As Mynamite points out in his/her answer, the first of the following pairs of sentences is correct; the second is not:

Me: "Life sucks!"

You: "Yeah, I agree with you. What is it all about?" (not "What is that all about?").

Here, the word that would be out of place.

On the other hand, let's change the dialog a little:

Me: "Life sucks!" (Then I start acting all depressed and down-in-the-dumps after having said what I said.)

You: "What's that all about?" (not "What's it all about?").

Here, the word that in your question is a pronoun that takes the place of all the things you've seen me doing since I said "Life sucks" (e.g., sulking, crying, complaining, whining, being critical, pouting, and so on). In this case, the word it would not be appropriate because I wouldn't know what it stands for (or takes the place of). I'd probably say, "What's what all about?"