Vai is used when you have a certain amount of choices and you have to pick one. Vai is only used in questions. (Usually foreigners tend to overuse this one. Soon you'll know better!)
- Otatko kahvia, teetä vai mehua? - Will you take coffee, tea or juice?
- Menisinkö elokuviin vai jäisinkö kotiin? - Should I go to the movies or should I stay at home?
Another common use of vai is vai mitä at the end of the sentence.
- Tavataan keskustassa, vai mitä? - Let's meet in downtown, right? (or what?)
Tai is used in commands and in affirmative sentences, but also in questions. In question, tai is used when there is a hidden idea of for example or or something else.
- Haluatko omenan tai appelsiinin? - Do you want an apple or an orange? (For example. Or something else to eat? You don't have to take anything.) Compare: Haluatko omenan vai appelsiinin? = I assume that you want either an apple or an orange, so now tell me which one.
- Osta maitoa tai piimää. - Buy milk or sourmilk/buttermilk.
- Soitan sinulle huomenna tai ylihuomenna. - I'll call you tomorrow or the day after tomorrow.
If you want to be really annoying in Finnish, end all your sentences with either vai mitä or tai jotain, which means or something.