What do we call the areas with "1" in a binary image
This problem born when you use pixel intensity as physical mass, for example to calculate mass center or inertia moments, in this case the background should be 0 and object should be 1.

Btw your question is not clear because in a binary image {0,1} have the same information content and generally there are not difference.

There is not a name for 1 pixel and 0 pixel, often in common convention 1 is white and 0 is black, but it's not a general rule. Maybe you can call 1 high value pixel and 0 low value pixel but I think that are not common names and they can make confusion in some contexts.