"¿Qué día es hoy?"
Translation:What day is it today?
In English, I would consider "What day is today" very formal, even stilted--the repetition of "day" sounds odd. In conversation, I would ask instead "What day is it?" Since absolute transliteration is not the goal here (and often marked incorrect in other cases, like "mucho gusto"), I think that answer should be at least an accepted alternative, if not the preferred translation.
"Que" is correct here because it is a general question. I just had a class on que vs. cual last night from a native speaker (with a phd in linguistics) and I would like to share what I learned. He mentioned that he often hears the incorrect use of que versus cual so he spelled out when they are correctly used. First, he said to forget that cual = which and que = what, and to focus instead on the circumstances when each is used. There are two situations.
The first situation depends on your expectation of the listener.
Cual is used when you anticipate the person has the answer to the question because everyone can answer the question. Examples: "Cual es su nombre?" or "Cual es su direccion?" In both cases you expect the person will know the answer because everyone will know the answer.
Que is used when you are saying "in case you have it, what is it?" For example, if someone has a new dog and you want to know if it has a name: "Que nombre tiene?" Another example would be "Que medicinas esta usando?" In the latter case you are asking what medicines a person is using, in case they are using any.
The second situation is when you have a general versus a specific question.
When you have a general question, use que, and when you have a specific question, use cual. To stick to an example from above, a general question would be "Que medicinas esta usando?" If the answer is "Estoy uso antibioticos." then the correct follow up question would be "Cual antibioticos?" Another example is "Cual es la diferencia?" because you are comparing two things, which is more specific.
For the example at issue, when you ask what day is it or what hour is it, you would use que because it is a general question (it could be any day or any hour). So you would say "A que hora es?" or "A que dia es?" Cual would be correct if you were asking a more specific question such as "Cual fue la hora de la primera cirugia?" ("What hour was the first surgery?") or "Cual dia es el concierto?" (What day is the concert?).
Hope that helps!
If only it were that simple. Here's a link that I hope helps.