Learn Russian: У студентки есть новая машина.

In Russian, you don't usually say "someone has something" directly. Instead, the phrasing translates closer to "by someone, there is something". The preposition У means "by" or "at", and it is followed by the owner in the Genitive case (студентки). The word есть means "is" or "exists". So literally, this sentence says "By the student there is a new car."

The dictionary word for female student is студентка. Why does it end in -и here instead of -ы?

The preposition У forces the noun into the Genitive case. Usually, feminine nouns ending in -а change their ending to -ы in the Genitive. However, Russian has a strict spelling rule: after the letter к (and a few other letters like г, х, ш, ж, ч, щ), you can never write the letter ы. You must write и instead. That is why студентка becomes студентки.

If the car is the thing being owned, why aren't новая машина in the Accusative object case?

Because of how the Russian "has" construction is built. Since the literal translation is "By the student there is a new car," the new car is actually the grammatical subject of the sentence. It is the thing doing the "existing." Because it is the subject, новая машина stays in the Nominative case, which is the basic dictionary form.

How would this sentence change if the student were male?