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The magnetic properties and phase diagrams of the mixed spin-1 and spin-1/2 Ising model on a checkerboard square structure have been studied using the Monte Carlo simulations based on the Metropolis update protocol. The system consists of four quartets of alternative spin configuration. The effect of the exchange interactions J and crystal field D on the magnetic properties, critical and compensation temperatures, susceptibility, and specific heat of the system have been investigated. The phase diagrams, T-J and T-D, for different values of the exchange interactions and crystal field have been examined. We found that the compensation temperature starts to evolve for J1 < 0.6 of the spin-1 assembly and for J2 > 1.6 for spin-1/2. On the other hand, the effect of ferrimagnetic coupling does not show any threshold value; in this case, the compensation temperature is constant. Regarding the crystal field strength, the threshold value of D > -0.8 has been observed. We obtain the N-, Q- and P-type compensation behaviors in the system. We have observed that the phase diagrams exhibit only a second-order phase transition to a paramagnetic phase; hence, the system does not show the tricritical point. The magnetic hysteresis cycles of the mixed spin-1 and spin-1/2 on a checkerboard square structure for different values of the exchange interactions, temperatures, and crystal fields have been found. Finally, the system exhibits the superparamagnetism behavior for a fixed value of the temperature and crystal field., Comment: 21 pages, 12 figures