We present new insights into the physical nature of coronal mass ejections (CMEs) and associated shock waves within the framework of the three-dimensional (3D) structure. We have developed a compound model in order to determine the 3D structure of multiple fronts composing a CME, using data sets taken from STEREO, SDO, and SOHO. We applied the method to time series observations of a CME on 2012 March 7. From the analyses, we revealed that a CME could consist of two different fronts: one is represented well with the ellipsoid model, implying that CMEs are bubble-shaped structures, and the other is reproduced well with the graduated cylindrical shell model, indicating that CMEs are flux rope-shaped structures. The bubble-shaped structure is seen as the outermost front of the CME, and the flux rope-shaped structure is seen as the bright frontal loop or three-part morphology. From our results, we conclude that (1) a CME could consist of two distinct structures, a bubble-shaped structure and a flux rope-shaped structure, (2) the bubble-shaped structure is a fast magnetosonic shock wave, while the flux rope-shaped structure is the mass carried outward by the underlying magnetic structure, (3) the driven shock front could be either a piston-shock type or a bow-shock type, (4) the observed EUV wave in the low corona is the footprint of the bubble-shaped wave, and (5) the halo CME is primarily the projection of the bubble-shaped shock wave but not the underlying flux rope.