Modern astrophysical simulation is the main aspect of human getting insight from the cosmos. Numerical simulations became irreplaceable tools for generating various phases of the universe, the formation of galaxies, and other astrophysical phenomena. As the advances of computational power, validity and diversity of astrophysical modeling have been growing. In addition to the development of the processor performance, the diversity of the software community is also rising therefore, research between various simulation codes is worthwhile. It allowed researchers to know the nature of different simulation techniques linked to their basic algorithms. This study summarizes the basics of common simulation methods used by researchers including N-body Simulations, Smoothed Particle Hydrodynamics, Monte Carlo Methods, Finite Difference and Finite Element Methods. Furthermore, this research analyzes the basics of AREPO, GIZMO, and GAMER-2 codes, and compared the performance of these codes. Last, according to the analysis, this study discusses and demonstrates the advantages and limitations of these codes. These results shed light on guiding further exploration of cosmology simulations.