The connection of distributed renewable energy and new re-electrified loads to the distribution network promotes the low-carbon production and consumption of energy. However, it also introduces a large number of source-load uncertainties and poses challenges to the safe operation of the distribution network. To quantitatively assess the impact of stochastic source-load, this paper proposes a theoretical framework of uncertainty quantification for distribution grid operation including uncertainty characterization and sensitivity analysis, starting from the modeling of source-load uncertainties in the distribution network. For uncertainty characterization, existing uncertainty characterization methods based on high-fidelity models and low-fidelity models are summarized in terms of computational efficiency and accuracy. Then, an uncertainty characterization method based on multi-fidelity models is proposed. For sensitivity analysis, global sensitivity analysis methods based on variance, area, and distance are introduced in this paper. Also, the characteristics of different sensitivity analysis methods are summarized and their applicability in various scenarios is compared. By exploring the application of the proposed uncertainty quantification theory in coupled systems, we aim to provide method support for uncertainty analysis of coupled systems in the multi-timescale operation.