This is how I was taught it (or understood I was taught it) - at law school, so it might have been dumbed down.

The impossibility is the impossibility of ensuring rational (transitive) outcomes amongst ranked preferences and adhering to a set of fair and democratic norms.

A rational transitive outcomes is one in which votes result in option A being preferred over option B and option B being preferred over option C, such that A is preferred over C (eg, A > B > C). Option A is known as the Condorcet winner.

But there may be cases where the vote yields no Condorcet winner (eg, A > B > C > A). This is illustrated by the following table:

Preferences Voter 1 Choc Vanil Strwb Voter 2 Vanil Strwb Choc Voter 3 Strwb Choc Vanil

Two voters prefer C over V and two prefer V over S, but two also prefer S over C.

To ensure transitivity, we can introduce voting rules, but it is impossible to introduce rules that do not violate the fair and democratic norms (referred to as the pre-specified criteria in the Wikipedia article: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives).