> a well-founded relation ⊏ is a well-ordering if and only if it relates every distinct pair of elements, i.e. x1≠x2 implies either x1⊏x2 or x2⊏x1.

The author here explains well-founded relation prior to this. I want to add an example here. Consider the set of positive real numbers. If we use the natural ordering < then we have a total order. But that's not a well-ordering. If x were to be the least element you could always divide it by 2 to get a lesser element. So there's no least element in the set of positive reals using the conventional < ordering. However, the amazing thing is that assuming axiom of choice, such a well ordering exists.