For voting systems, the criterion can be illustrated by its use in the more specific Gibbard–Satterthwaite theorem.
Every non-dictatorial, non-imposing voting system for choosing a single candidate from more than two candidates is susceptible to tactical voting, i.e., voting against one’s preferences to effect one’s preferences.
Now, the boldfaced parts are the main insight to be had here for real elections.
Still, the conditions non-dictatorial and non-imposing are necessary to cover the following pathological cases:
A system where the choice of a single voter (the dictator) is given absolute preference, i.e., whoever that dictator chooses wins irrespective of the other votes.
A system where some candidates cannot win irrespective of the votes, e.g., the system where the tallest candidate automatically wins. (The outcome is imposed by the voting system and not affected by the voters at all.)
In the first system, the dictator cannot vote tactically since the outcome is identical to their vote. Everybody else in those two cases cannot vote tactically since their vote has no effect at all. So, both systems are immune to tactical voting and thus the theorem would be wrong if it did not impose the respective conditions. Of course, the immunity to tactical voting does not outweigh the fact that those systems are so absurd that not even tinpot dictators would try to market them as fair voting systems.
The property of non-dictatorship is satisfied if there is no single voter i with the individual preference order P, such that P is the societal ("winning") preference order, unless all voters have the same P. Thus, as long as there are voters in the society that have different preference orderings, the preferences of individual i should not always prevail.
Like David Richerby, I find this formulation misleading. In particular the restriction “unless all voters have the same P” does not make sense to me in any interpretation of those words. Here is Arrow’s definition from A difficulty in the concept of social welfare:
In its pure form [dictatorship] means that social choices are to be based solely on the preferences of one man. That is whenever the dictator prefers x to y, so does society. […]
Definition 5: A social welfare function is said to be “dictatorial” if there exists an individual i such that for all x and y, xPiy implies xPy regardless of the orderings of all individuals other than i, where P is the social preference relation corresponding to those orderings.
Translated to the context of elections, “social welfare function” corresponds to the voting system, “social preference relation” corresponds to the voting outcome, and xPiy or xPy means that Pi or P prefers x to y.
No special attention is paid to the case that all voters have the same preferences. Instead the key feature of dictatorship is that the dictator’s preference prevail for all possible preferences of the other voters.