"These old buildings are banks."
Translation:Ezek a régi épületek bankok.
Öreg for people or creatures or other living things, if you're talking about their age (you could even talk about an öreg fa, an old tree) ; régi for inanimate things. Régi is closer to "outdated" or "previous" or "from an earlier time." So you could talk about a régi barátnőm - my old girlfriend, as in the girlfriend I used to have. That is quite different from az öreg barátnőm, my elderly girlfriend.
What's the difference between öreg and idős, then?
Very little, I think. Only about as much as the difference between "old" and "elderly".
The adjectives ez and az (and ezek and azok in the plural) are always connected to their nouns with the definite article a(z).
a bank - the bank
ez a bank - this bank
az a bank - that bank
ezek a bankok - these banks
az asztal - the table
ez az asztal - this table
az az asztal - that table
azok az asztalok - those tables