"Is that cheese or milk?"
Translation:Onko tuo juustoa vai maitoa?
This is the partitive case. Finnish has a load of different cases for nouns to be in. Partitive is used when something has an indefinite amount or can't be counted. It's kind of similar to saying "any" or "some" in English. The partitive case is relative simple, in most instances its simply noun + a.
For example: "Minulla on ketsuppia" - I have some ketchup. "Minulla on ketsuppi" - I have (the) ketchup.
The tips pages have more information on how these work and other instances in which the partitive case applies: duolingo.com/skill/fi/Fridge/tips-and-notes
Simply put, "vai" is used when choosing between particular options, and "tai" is more uncertain and question-like (options aren't necessarily fixed/limited).
E.g. "Do you want green or black tea?" / "Haluatko vihreää vai mustaa teetä?"; you can choose either one or the other. Maybe those are all they're serving in general, or you have already defined that your choice of beverage is tea. This or that, we know they have both.
"Do you have coffee or juice?" / "Onko teillä kahvia tai mehua?"; they might or might not have one or both, and the possibility for other options isn't limited.
"Haluatko ketsuppia tai sinappia?" - Do you want ketchup or mustard? -> you can choose one, both or neither
"Haluatko ketsuppia vai sinappia?" - Do you want ketchup or mustard? -> you can choose only one of these, and it is assumed that you are going to choose, i.e. that you are not going to say "neither"