This work explores the optical properties of a photonic crystal ring resonator (PhCRR), a device consisting of a microring resonator upon which a photonic crystal structure is superimposed.  Due to the periodic dielectric structure of the PhCRR, the gradient of the device's dispersion curve approaches zero near the photonic band edge, resulting in enhanced light-matter coupling and quality factors due to the low group velocity of the resonant modes. In order to fully exploit the ``slow light" characteristics of the PhCRR, a design approach is used which allows for the selection of band edge resonant modes. A frequency domain computational approach models the dispersion of a periodic silicon photonic crystal waveguide. Boundary conditions are then imposed on the waveguide, ensuring the phase matching of propagating electromagnetic waves and the discreteness of the number of lattice cells in the ring. Through proper selection of design parameters, these geometric constraints return a set of resonant modes which fall precisely at the photonic band edge. Finite-difference time-domain simulations yield the field energy densities of the individual resonant modes of the PhCRR, with calculated quality factors greater than 10^7.  The spectral features of the PhCRR and the effect of geometric disorder are explored.  Finally, a design proposal for the silicon-on-insulator fabrication of on-chip photonic crystal ring resonators is discussed.