The Geometrically Intrinsic Nonlinear Recursive Filter, or GI Filter, is designed to estimate an arbitrary continuous-time Markov diffusion process X subject to nonlinear discrete-time observations. The GI Filter is fundamentally different from the much-used Extended Kalman Filter, and its second-order variants, even in the simplest nonlinear case, in that: * It uses a quadratic function of a vector observation to update the state, instead of the linear function used by the EKF. * It is based on deeper geometric principles, which make the GI Filter coordinate-invariant. This implies, for example, that if a linear system were subjected to a nonlinear transformation f of the state-space and analyzed using the GI Filter, the resulting state estimates and conditional variances would be the push-forward under f of the Kalman Filter estimates for the untransformed system -- a property which is not shared by the EKF or its second-order variants. The noise covariance of X and the observation covariance themselves induce geometries on state space and observation space, respectively, and associated canonical connections. A sequel to this paper develops stochastic differential geometry results -- based on ``intrinsic location parameters,'' a notion derived from the heat flow of harmonic mappings -- from which we derive the coordinate-free filter update formula. The present article presents the algorithm with reference to a specific example -- the problem of tracking and intercepting a target, using sensors based on a moving missile.