I know there is fiat currency, usually printed up by a central bank on paper. There is also commodity currency (gold, silver, etc).
Are these the only kinds of currency, or are there others?
fiat currency, just means money by political fiat. so it can be commodity or fiduciary in theory, though in practice when you see it in the world expect it to be fiduciary fiat money.
Another point is the distinction between debt-based fiat money (borrowed from a central bank) and credit-based fiat money (directly spent into circulation by the government).