The intersection-button constructs the intersection of two selected objects or asks to select two arbitrary objects.
For almost all pairs of objects where this makes sense at all, an intersection is defined. The most important exception are intersections with parallelograms and triangles: Those are defined for other parallelograms and triangles, planes, lines and segments only. For instance the intersection of a triangle with a sphere are one up to three circle-segments, which are not yet implemented in Archimedes.
If one or both of the selected objects are groups, all grouped objects will be intersected with the other object. This way one can easily construct intersection-polygons of platonic solids with planes.