Friction exists in most mechanical systems, and it can have a major influence on their dynamic performance and operating conditions. As a consequence of frictional contact phenomena, energy is dissipated and the state of a system can change slowly and rapidly, depending on the nature of the contact, continuous or impact condition. Other effects associated with friction in mechanical systems are the vibration and noise propagation of the system components, nonlinear systems’ behavior and wear. Overall, the knowledge of the friction regimen, as well as the frictional forces developed at the interface of mechanical parts in contact with relative motion, is crucial for the dynamic analysis of mechanical systems, and has consequences in the design process. Thus, this work is a review of the modeling and analysis of frictional effects in multibody systems with the purpose of better understanding and obtaining accurate responses. In this process, pure dry sliding friction, stick–slip effect, viscous friction, Stribeck effect, and frictional lag are some of the main phenomena associated with friction, which are addressed in depth. Overall, the friction models can be divided into two main groups, namely the “static friction models” and the “dynamic friction models”. The static models describe the steady-state behavior of the relation friction-force/relative-velocity, while the dynamic models allow for the capturing of more physical responses and properties by using extra state variables. In a simpler manner, the static and dynamic friction models differ mostly in the modeled frictional effects, implementation complexity, and computational efficiency. Hence, this research is aimed at analyzing in detail the role of friction modeling in the dynamic response of multibody system, as well as addressing the importance of friction models selection for accurately describing the friction related phenomena. Demonstrative application examples, which include friction in ideal mechanical joints and systems involving contact–impact events, including an example of rolling contact will be considered and investigated to illustrate the main assumptions and procedures adopted in this work. The results from this study indicate that in most cases, a static friction model, which accounts for static friction and avoids the discontinuity at zero velocity, is a suitable choice. A more advanced dynamic friction model has to be developed to be utilized for systems containing high variations of normal load, namely with impact conditions.