A reconfigurable intelligent surface (RIS) assisted scheme is proposed for direction-of-arrival (DOA) estimation in this work. In the scheme, the RIS is utilized to provide a reflected path, and the RIS-reflected signal is received by array antennas together with the direct-path signal. The model consisting of the direct-path signal and the RIS-reflected signal is proposed, and the Cramer-Rao Bound (CRB) is derived based on the signal model to assess the ultimate estimation performance. In order to enhance the estimation performance of RIS-assisted scheme, two possible strategies for the phases design of RIS are presented. The first possibility considers RIS phase shifts that maximize the coherence of signals from the direct path and the RIS-reflected path. The other possibility considers to obtain the phase shifts that minimize the CRB of DOA estimation in the proposed scheme through a manifold optimization method. Moreover, DOA estimation methods are proposed for the RIS-assisted scheme to validate the scheme feasibility. Target tracking is proposed as a practical application for the RIS-assisted scheme. In the end, both theoretical analysis and simulation results illustrate that the accuracy of DOA estimation and target tracking has been improved with the aid of RIS.