Learn Russian: Чья там новая машина?

Чья means "whose", but in Russian, this question word acts like an adjective. It has to match the gender of the noun it belongs to. Since машина (car) is a feminine noun, we use the feminine form чья instead of the masculine form чей.

Why does the adjective новая end in -ая?

Because it is describing машина, which is a feminine noun. Adjectives must match the gender of the noun they describe, so the standard masculine dictionary form новый changes to the feminine новая. Notice how чья, новая, and машина all end in -а or -я — this rhyming pattern is a great hint that they are all feminine and belong together!

Why does там (over there) come right after the question word? Can I put it at the end?

Yes, you can! Russian word order is very flexible, so you could perfectly well say Чья новая машина там? However, putting там early in the sentence is a very natural way to speak. It quickly points out where you are looking before giving the details about the new car.

Where is the word "is" in this Russian sentence?

In Russian, the present tense of the verb "to be" (like "is", "am", or "are") is almost always skipped. The sentence literally translates to "Whose there new car?", which we automatically understand as "Whose new car is over there?".