Breathing room in a composition is the space between the subject and the top and bottom edges of your square/rectangle, as well as the left and right-hand sides. In most cases, you will want to leave enough room so that your subject doesn't feel restrained by placing it too close to an edge.
For example, in the painting below by George Angelini, notice how the primary element (the woman in the chair) has enough space on the left-hand side of the frame so that the viewer doesn't get the feeling of being claustrophobic. This effect is achieved by placing the subject's back closer to the right-hand edge of the rectangle, and therefore, almost doubling the space on the left-hand side.
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