The Dirac equation plays a fundamental role in relativistic quantum mechanics and its solution is great interest for the theoretical understanding of broad class of problems in the theory of elementary particles. Studying of bound states gives us a lot of information about the nature of the system. An outstanding goal is not to discuss Quantum Field Theory and Quantum Electrodynamics in this book. However, we shall discuss the Dirac equation which reduces to four coupled partial differential equation, which describes the relativistic electron and other spin half particles. In addition to this, the aims of this book is to evaluate and study the bound state of the relativistic hydrogen like atom, Dirac particle in a spherical attractive central square well potential. Finally, we study the solution of the relativistic Dirac equation for a Yukawa-type potential and compare with the non-relativistic Schrodinger spectrum of the energy level for hydrogen atom.