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In this paper the linear and nonlinear theory of the electron cyclotron maser instability is considered. The configuration used to study the laser instability consists of relativistic electrons gyrating about and drifting along a uniform magnetic field within a parallel plate waveguide. Relativistic effects associated with the gyrating electrons are responsible for excitation of the transverse electric mode in the waveguide. Linear theory shows that the growth rate maximizes when the actual beam velocity coincides with the axial wave group velocity of the excited electromagnetic wave. This allows the performance of nonlinear analysis in a frame where both the axial wave number and axial beam velocity vanish. The nonlinear analysis shows that there are two possible mechanisms for the saturation of the unstable wave: i. depletion of the available free energy associated with the rotating particles. ii. phase trapping of the gyrating electrons in the wave. The initial beam parameters determine which of the two mechanisms is responsible for saturation. Competition between the two saturation mechanisms leads to a peaking in the energy conversion efficiency as a function of beam energy. Numerical results of the nonlinear formalism show that energy conversion efficiencies from the particles to the wave can be as high as 60 percent in the beam frame.