If I have 4 routers, R1, R2, R3, and R4, and put all of the subnets from R1 and R2 in area 0, and put all of the subnets from R3 and R4 (including the routers) into OSPF area 1, would the ospf areas be shared such that OSPF on router R1 corresponds to OSPF on R2, and same with R3 and R4? And if so, how do you communicate between areas?
OSPF is arranged in areas, and all inter-area traffic must pass through the backbone area (Area 0). A router with interface(s) in Area 0 and a different area is an ABR (Area Border Router), through which traffic can pass between areas. You cannot, for example, have a router with one interface in Area 1, and another interface in Area 2 and expect it to pass traffic between those two areas; all inter-area traffic must pass through Area 0. This helps to prevent routing loops.
The OSPF routers in a particular area have a full understanding of all the routes, paths, and routers in that area. An ABR, as a router in a non-backbone area and Area 0, has that knowledge of both areas.
I suppose a good analogy is to think of a businesses private network as an OSPF area, and then on the exterior a business will have a DMZ area which allows connections to the world-wide Internet (another area).
Thus, each of your networks can only communicate with the "outside world" from its edge (like the DMZ). That's one reason you're being asked for the topology: how specifically do the parts of your net relate to one another?