I've seen a few mansions designed so that the house is a sort of square where the center part of the square shaped mansion/house contains an outside area.
If the area is uncovered, open to the elements, it's a courtyard.
An unroofed area that is completely or partially enclosed by walls or buildings, typically one forming part of a castle or large house.
According to some definitions, atrium might be appropriate, but I agree with the following:
The central uncovered area in a Roman domus was referred to as an atrium. Today, we generally use the term courtyard to refer to such an area, reserving the word atrium to describe a glass-covered courtyard.
This can be called an atrium.
1. Architecture A rectangular court, as: a. A usually skylit central area, often containing plants, in some modern buildings, especially of a public or commercial nature. b. The open area in the center of an ancient Roman house.
As you can see, this is derived from the layout of Roman houses, which featured just such a central open area.
According to the OED an "atrium" is usually a covered area, though often with skylight of some kind. The closest to what @Robusto describes is sense 1d. d. In a public building, a usually skylit central court rising through several storeys and surrounded by galleries at each level with rooms (shops, offices, etc.) opening off them. orig. U.S.
A building which has an especially splendid interior court is the Foreign Office in London, they call it "the Quadrangle".
The term atrium is used in Britain, but mostly for an enclosed space in a modern building. In an older building it would normally be called a Quadrangle.
The ancient Romans called this an atrium, and as far as I know this word is still used today, both for large buildings like hotels and shopping centers, as for smaller ones like houses:
2 a: a rectangular open patio around which a house is built
A non-Latin term for this is simply 'indoor garden'.