Alum isn't a specific chemical name, it's sort of like "nitrate" or "hydrate" in that it specifies a chemical group that is attached to something else. And, it's an oldie name, not a standard one, so it isn't as descriptive as it might be.
Cake alum, the kind in the grocery store is:
Aluminum sulfate Al2(SO4)3
Aluminum is a metal, aluminum sulfate is aluminum bonded with sulfate groups. The two groups should disassociate in water, adding ions of aluminum and sulfate to the solution.
There are other types of alum too.