Auger recombination is an important nonradiative carrier recombination mechanism in many classes of optoelectronic devices. The microscopic Auger processes can be either direct or indirect, mediated by an additional scattering mechanism such as the electron-phonon interaction and alloy disorder scattering. Indirect Auger recombination is particularly strong in nitride materials and affects the efficiency of nitride optoelectronic devices at high powers. Here, we present a first-principles computational formalism for the study of direct and indirect Auger recombination in direct-band-gap semiconductors and apply it to the case of nitride materials. We show that direct Auger recombination is weak in the nitrides and cannot account for experimental measurements. On the other hand, carrier scattering by phonons and alloy disorder enables indirect Auger processes that can explain the observed loss in devices. We analyze the dominant phonon contributions to the Auger recombination rate and the influence of temperature and strain on the values of the Auger coefficients. Auger processes assisted by charged-defect scattering are much weaker than the phonon-assisted ones for realistic defect densities and not important for the device performance. The computational formalism is general and can be applied to the calculation of the Auger coefficient in other classes of optoelectronic materials.