I think “democracy” is the worst-understood concept in politics. Or, rather, while there may be more confused notions—capitalism comes to mind—people usually have some awareness of the issues’ complexities and are at least a bit wary of invoking their names as absolutes. Statements like “mandatory health care is anti-capitalist” provoke some minimum amount of reflection on the meanings of the words used; reaction to statements like “judicial review is undemocratic” tends to skip right past such parsing and on to consequentialist and historical argument.
The only widely-agreed definition is that in order for something to be democratic it must derive legitimacy from individual decision-making by all members of a group. It’s crucial to note that this description addresses the adjective and not the noun. It’s very common to use “democracy” as a shorthand for “a democratic political/governmental system”, but I think this is the start of much of the confusion: systems are not monolithic. In particular, decisions over how a political system is organized (its constitutional structure) and decisions made in the course of its functioning (e.g. its legal code) need not be arrived at under the same framework. The structure of a democratic process can be mandated undemocratically—a king can decree that peasants vote on which crown he should wear.
The real trouble is that people conflate the process of democratic decision-making with larger sociological or epistemic notions. Making a decision according to some codified set of rules doesn’t mean that people “agree”. It doesn’t mean the decision is moral or practical or that its suppositions are true. And despite the underlying goals of democratic organization, it doesn’t even guarantee that all members of the group will be satisfied with or respect the decision. A process is just a process, and wrapping decision-making up in constitutional structure based on enlightenment ideals doesn’t change that.
Any time a decision is made that will affect many people, whether by a king or a representative body or by a referendum of all members, and whether agreed by 51% or 67% or 90%, the various impacts on different subgroups need to be considered by the people making that decision. One thousand indifferent “yea”s may carry more constitutional power than a single well-considered and forceful “nay”; that doesn’t give them moral authority.
Democracy doesn’t absolve voters of responsibility any more than capitalism obviates altruism or the legal system obviates ethics. Setting aside differences in service of a greater purpose is not an emergent property of democracies; it is a prerequisite that must be met by each individual.