It looks to be a double boiler of some sort, but we're not sure what it is.
It's an egg poacher. Looks like this exact model.
Note that these don't really actually "poach" eggs, the result is more like a coddled egg. They do essentially work like double boilers.
If you put a dab of butter in the bottom of the cup and and a raw egg over it, then steam it, you'll have coddled or shired eggs.