The theoretical work of V.B. Braginsky predicted that radiation pressure can couple the mechanical, mirror-eigenmodes of a Fabry-Perot resonator to its optical modes, leading to a parametric oscillation instability. This regime is characterized by regenerative mechanical oscillation of the mechanical mirror eigenmodes. We have recently observed the excitation of mechanical modes in an ultra-high-Q optical microcavity. Here, we present a detailed experimental analysis of this effect and demonstrate that radiation pressure is the excitation mechanism of the observed mechanical oscillations.