The different forms of jurisdictions for the Supreme Court help by hearing non-biased cases in court. Under the original jurisdictions for the Supreme Court, or the first court to hear the case, some of the cases that can be heard are:
Disputes between two states
Disputes where the United States is a party to the case
Disputes between individuals who do not share a common state
Habeas Corpus (or false imprisonment)
Bankruptcy
I hope this helps. There are more, but those are a few that I know of.