From Gerald Erichsen,
Your Guide to Spanish Language.

Qué and cuál form one of those word pairs that can be confusing to people learning Spanish, because they both are often translated as the same words into English.
The two words, when they have accents, are used as pronouns in questions to mean "what" or "which." (Without accents, they normally aren't used in questions and most often are connecting words or, technically speaking, relative pronouns, translated "which" or "that." Those forms will be explained in a future feature.) They can stand for either feminine or masculine objects as well as ideas or concepts. Qué is the same in the singular and plural forms; the plural of cuál is cuáles.

It is sometimes said that qué is closer to "what" in meaning, and cuál is closer to "which." But that rule isn't always reliable.

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Here is a guide to knowing which of the two pronouns to use:

Qué is used when asking for definitions.

¿Qué es una ciudad? What is a city?
¿Qué hace un presidente? What does a president do?
¿Qué significa "talanquera"? What does "talanquera" mean?

Qué normally is most often used before nouns (although cuál isn't necessarily incorrect and you will hear it sometimes used to mean "which"):

¿De qué color es la camisa? What color is the shirt?
¿Qué casa prefieres? Which house do you prefer?
¿Qué libro leíste? Which book did you read?

Cuál usually is used before es and other forms of ser when not seeking a definition:

¿Cuál es tu número de teléfono? What is your phone number?
¿Cuál es tu problema? What is your problem?
¿Cuáles son la ciudades más grandes? What are the biggest cities?

Cuál is used to suggest a selection or choice from among a group:

¿Cuál miras? Which one are you looking at? But, ¿Qué miras? What are you looking at?
¿Cuál prefieres? Which one do you prefer? But, ¿Qué prefieres? What do you prefer?
¿Cuáles quieres? Which ones do you want? But, ¿Qué quieres? What do you want?
¿Cuál es el más caro? Which is the most expensive?
¿Cuáles son los más caros? Which are the most expensive?

Qué has some common idiomatic uses:

¿Qué hora es? What time is it?
¡Qué lástima! What a shame! ¡Qué hermoso! How beautiful! ¡Qué día más horrible! What a horrible day! ¡Qué susto! What a fright! ¡Qué bonito! How pretty!
¿Qué tanto? How much?
¿Y qué? ¿Y a mí qué? So what?
¿Para qué? ¿Por qué? Why?
Note that, which the exception of the qué exclamations, the accented forms of these words appear only in questions. Note also that it is most common to find these words at the beginning of a sentence, although qué sometimes follows de and other prepositions