The three-dimensional flow characteristics of the compressible vortex ring generated by under-expanded circular jets with two typical pressure ratios, i.e. $n=1.4$ (moderate) and 4.0 (high), are investigated numerically with the use of large-eddy simulations. Our results illustrate that these two pressure ratios correspond to different shock structures (shock cell and Mach disc, respectively) within the jet. These two typical types of flow structures and characteristics are discussed and validated with experiments, and the different generation mechanisms of the secondary vortex rings are compared. Moreover, detailed information about the evolution of the secondary vortex ring, primary vortex ring and turbulence transition features, including the radial and azimuthal modes, is investigated. The geometric features and mixing effects of the jets are also explored.