Learn Italian: Di chi è quella macchina?

Italian doesn't have a single exact word for "whose". Because di shows possession, you ask "whose" by saying Di chi (literally "Of whom"). You are essentially asking "Of whom is that car?".

Why is the word order so different from English?

In English we group the question word and the object together: "Whose car...". In Italian, you start with Di chi and usually follow it immediately with the verb, which is è (is) here. The thing being owned (quella macchina) goes at the end of the sentence.

Why do we use quella instead of just quel?

Quella is the feminine singular form of the word for "that". It has to match the noun it describes. Since macchina (car) is a feminine word, we use quella macchina. If it were a masculine word like libro (book), you would use quel libro.

Could I say Di chi è la macchina? instead?

Absolutely! That means "Whose car is it?" or "Whose is the car?". Using quella simply changes the meaning to "Whose car is that?", pointing out a specific car.