Shock wave is a common phenomenon in astrophysics. Shock wave acceleration has been regarded as a source of high-energy cosmic rays. Very strong magnetic field exists in the surrounding of the shock wave at the edge of the supernova remnants. But the mechanisms of generation and amplification of such a strong magnetic field are not clear yet. In this paper, the properties of shock wave driven by the laser irradiating on un-magnetized and magnetized plasmas are investigated using two-dimensional particle-in-cell (PIC) simulations. It is found that very strong spontaneous magnetic field can be generated around the laser-driven shock front in the un-magnetized plasma. The spontaneous magnetic field can store energy and accelerate electrons further. When an external magnetic field is introduced, the electrons and ions are accelerated more efficiently by the shock wave than in the un-magnetized plasma. The external magnetic field can transfer its energy to electrons and ions, and strengthen the shock wave. In simulations, the introduced external magnetic field has three different strengths: 1072 MG, 107.2 MG and 10.72 MG, which determine the shock structures through the driven currents. There are two single-polar magnetic arcs that constitute the shock structure when the external magnetic field is 1072 MG, i.e., one is the shock itself and the other is actually the reverse shock, whereas only one magnetic arc is produced but with a bipolar structure in the direction perpendicular to the shock propagation when the externally added magnetic fields are much lower (107.2 MG and 10.72 MG). The two bipolar magnetic structures will evolve into a single-polar arc when the externally added magnetic field is 107.2 MG, but they are kept for all the time when the external magnetic field is 10.72 MG. It can be explained by taking the Larmor radius into the consideration. That the amplification ratio of the magnetic field decreases as the introduced external magnetic field increases implies that the magnetic amplification in the space is possibly due to the local field generation rather than the field compression. An amplification ratio of tens of the external magnetic field is achieved due to the pseudo Rayleigh-Taylor instability, but still much smaller than that around the astrophysical shock front, indicating that other efficient mechanisms are responsible for the observed magnetic amplification around shocks in the supernova remnants.