Measures of two types of uncertainty that coexist in the Dempster-Shafer theory are overivewed. A measure of one type of uncertainty, which expresses nonspecificity of evidential claims, is well justified on both intuitive and mathermatical grounds. Proposed measures of the other types of uncertainty, which attempt to capture conflicts among evidential claims, are shown to have some deficiencies. To alleviate these deficiencies, a new measure is proposed. This measure, which is called a measure of discord, is not only satisfactory on intuitive grounds, but has alos desirable mathematical properties. A measure of total uncertainty, which is defined as the sum of nonspecificity and discord, is also discussed. The paper focuses on conceptual issues. Mathematical properties of the measure of idscord are only stated ; their proofs are given in a companion paper