Transitivity is a property of relations. Unlike a lot of other mathematical language, it's very intuitive and easy to grasp, so I don't feel bad for writing it in hard math form.

A relation between two sets, A and B, is a set of ordered pairs whose first elements are in A and whose second elements are in B. So the equality relation on the natural numbers looks like {(0, 0), (1, 1), (2, 2), ...}. Or I could form a relation of the Presidents and their years of inauguration like so: {(George Washington, 1789), (John Adams, 1798), ...}.

Okay, that's pretty basic. Well, a relation from a set to itself has the transitive property if, whenever aRb and bRc (that means (a,b) and (b,c) are in the relation), we also know aRc. We want this property in our very nicest classes of relations: equivalence relations, which must also be reflexive and symmetric, and orders, which must also be antisymmetric. (Weak orders are reflexive, strict orders are irreflexive.)

A lot of the real life relations we use are transitive. "Weighs more than" or "is taller than" are good examples, and note that we can use these to order people. Douglas Hofstadter had an example in which the relation was "wrote a book, one of whose characters was." On the equivalence relation side, how about "is in the same family as"? On the other hand, a lot of the problems with running tournaments come from attempting to rank people (or teams) with the relation "could beat," which decidedly isn't transitive.