Learn Finnish: Onko tämä mehua vai viiniä?

In Finnish, mass nouns—things you cannot easily count, like liquids, food, or materials—are put in the partitive case when talking about an undefined amount. Since you are asking if the liquid is juice or wine in general, you use the partitive endings -a and -ä.

Why does mehua get an -a ending, but viiniä gets an -ä ending?

This happens because of vowel harmony. The word mehu has the back vowel u, so its ending must use the back vowel -a. The word viini only contains the neutral vowel i, and words with only neutral vowels take front vowel endings like -ä.

I have learned that tai means "or". Why does this sentence use vai?

In Finnish, vai is used for "or" in questions where the listener has to choose exactly one of the options. You use tai in regular statements, or in questions where the answer could be "both" or "neither".

How does onko make this a question?

The word onko is the verb on (is) with the question particle -ko attached to it. Adding -ko (or -kö) to a word and putting it at the start of the sentence is the standard way to form a yes-or-no question in Finnish.