Natural or induced fractures are typically present in subsurface geological formations. Therefore, they need to be carefully studied for reliable estimation of the long-term carbon dioxide storage. Instinctively, flow-conductive fractures may undermine storage security as they increase the risk of CO2 leakage if they intersect the CO2 plume. In addition, fractures may act as flow barriers, causing significant pressure gradients over relatively small regions near fractures. Nevertheless, despite their high sensitivities, the impact of fractures on the full-cycle storage process has not been fully quantified and understood. In this study, a numerical model is developed and applied to analyze the role of discrete fractures on the flow and transport mechanism of CO2 plumes in simple and complex fracture geometries. A unified framework is developed to model the essential hydrogeological trapping mechanisms. Importantly, the projection-based embedded discrete fracture model is incorporated into the framework to describe fractures with varying conductivities. Impacts of fracture location, inclination angle, and fracture-matrix permeability ratio are systemically studied for a single fracture system. Moreover, the interplay between viscous and gravity forces in such fractured systems is analyzed. Results indicate that the fracture exhibits differing effects regarding different trapping mechanisms. Generally speaking, highly-conductive fractures facilitate dissolution trapping while weakening residual trapping, and flow barriers can assist dissolution trapping for systems with a relatively low gravity number. The findings from the test cases for single fracture geometries are found applicable to a larger-scale domain with complex fracture networks. This indicates the scalability of the study for field-relevant applications.