Learn French: Quel manteau est-ce que vous voulez ?

Quel is an interrogative adjective, which means it must agree in gender and number with the noun it describes. Because manteau (coat) is a masculine, singular noun, we use the masculine singular form quel. If you were asking about a feminine noun like chemise (shirt), you would use quelle chemise.

Does quel mean "which" or "what" in this sentence?

Here it translates to "which", because the speaker is asking the listener to choose a specific coat from an available selection. Quel can mean either "which" or "what" depending on the situation, but before a noun where a choice is implied, "which" is the most natural English translation.

Why is est-ce que placed right after Quel manteau?

When you form a question using a question phrase like Quel manteau, that phrase goes at the very beginning. You then add est-ce que to signal that a question is coming, followed by the normal subject-verb order: vous voulez. This is a very common and natural way to ask questions in everyday French.

How would this sentence change if I wanted to ask "Which coats do you want?"

You would need to make both the noun and the question word plural. The noun manteau takes an -x to become manteaux, and quel takes an -s to become quels. The sentence becomes: Quels manteaux est-ce que vous voulez ? Keep in mind that because those added final letters are silent, the plural sentence sounds exactly the same as the singular sentence when spoken!