The development of numerical N-body simulations have allowed studying the formation process and evolution of galaxies at different scales. This paper presents the fundamental concepts of N-body systems applied to the cosmological evolution of the ¤-Cold Dark Matter (¤CDM) model. To perform structure formation in the Universe, we provide an introduction to the basic equations and their implementation on the GADGET-2 software. We also present a simple guide to modifying this code. First, we briefly describe the dark matter in the Universe as well as the theoretical and experimental basis of the ¤CDM model. Then, we focus on the simulation codes and provide the equations that govern most of the N-body simulations to model the dark matter. We describe the Smoothed Particle Hydrodynamics method used for simulating the gas, star dynamics, and structure formation in these simulations. Then, cautiously, we guide the reader to the installation of GADGET-2 on a Linux-based computer, as well as to carry out a couple of examples to operate the code. Finally, by using a computational cluster, we show several results of a large structure simulation, analyze the outputs to display the matter power spectrum, and compare the outcome with theoretical predictions.