It's a grammatical case. The accusative case is used for the object of the sentence. The nominative case is used for the subject. An example of this is "I greet him". I am the subject, so the nominative "I" is used. He is the object, so the accusative "him" is used. The sentence "he greets me" instead has the nominative third person singular "he", and the accusative first person singular "me". Hope that helps.
Either one is possible. But it's important to remember the two aren't interchangeable. "Du/dig" are for when you're talking to a single person, "I/jer" is for when you're talking to more than one person (or to someone from a group of people, but including everyone else in that group) .
Is fortaeller ever used outside of this "fortaeller ... om" construct? Also, does om not always mean about?
I'm asking because it seems unnecessary to teach this verb as "fortaeller ... om". It seems like you could just teach the words individually, and then put this together. That they didn't take that approach makes me think that something must be going on here, but I can't figure out what it is.