We present a methodology for calculating the electromagnetic radiation from accelerated charged particles. Our formulation-the "endpoint formulation"-combines numerous results developed in the literature in relation to radiation arising from particle acceleration using a complete, and completely general, treatment. We do this by describing particle motion via a series of discrete, instantaneous acceleration events, or "endpoints," with each such event being treated as a source of emission. This method implicitly allows for particle creation and destruction, and is suited to direct numerical implementation in either the time or frequency domains. In this paper we demonstrate the complete generality of our method for calculating the radiated field from charged particle acceleration, and show how it reduces to the classical named radiation processes such as synchrotron, Tamm's description of Vavilov-Cherenkov, and transition radiation under appropriate limits. Using this formulation, we are immediately able to answer outstanding questions regarding the phenomenology of radio emission from ultra-high-energy particle interactions in both the earth's atmosphere and the moon. In particular, our formulation makes it apparent that the dominant emission component of the Askaryan effect (coherent radio-wave radiation from high-energy particle cascades in dense media) comes from coherent "bremsstrahlung" from particle acceleration, rather than coherent Vavilov-Cherenkov radiation.