Aluminium and copper are widely used in engineering structures, due to their unique performances, such as higher electrical conductivity, heat conductivity, corrosion resistance and mechanical properties even though they have considerable differences in their melting points. In this study, the microstructure of the friction stir spot welds of aluminium and copper produced at various parameter combinations were analyzed by using a scanning electron microscope; while the residual stresses were studied by using the X-ray diffraction technique. Furthermore, the electrical resistivities of the joints was also measured. The evolving microstructure shows a good mixing in the produced spot welds with Cu particles present in the aluminium matrix. The formation of a copper ring/hook was evident in all the spot welds; and the length thereof increased with the shoulder plunge depth variation; while the spot welds produced at 1200 rpm for the two tool geometries exhibited a decrease and a slight increment in the length of the copper ring using a flat pin/flat shoulder and conical pin/concave shoulder, respectively. The obtained residual stresses results were compressive. The maximum residual stress of −116.8 MPa was measured on the copper ring of the welds produced at 800 rpm and 0.5 mm shoulder plunge depth, when using a flat pin and a flat shoulder tool. This was due to the generation of stress, when the copper was extruded into the aluminium sheets. Furthermore, the intensity of all the peaks using different process parameters decreased in comparison to the peaks generated by the parent materials and the effect of shoulder plunge depth on the full width at half the maximum (FWHM) was observed. The values of the measured electrical resistivities of the joints were higher than those of the parent materials.