The standard definition of transversality is that it describes a condition imposed within an economic model.
The model here is one that involves choice over an infinite period. The transversality condition is said to rule out choices that involve infinite debt or other implausible results (About.com).
Transversality is fundamentally an end-point constraint, where the “end-point” is the last “T” (for time) after an infinity of T’s. The end-point constraint is there because of the temporary violation of the familiar “budget constraint” in consumer theory, where the actor has a limited budget with which to make purchases for consumption. When the time period is infinite, the transversality condition comes in to keep such actors “honest” since, of course, a borrower should repay. When such actors are in fact dishonest, transversality is violated.
How does transversality relate to present macro models?
Theorists claim probity of their model in part because they impose transversality conditions. There is nothing apparently wrong with that, except when there is irrationality, market bubbles, Ponzi schemes, etc. Such events, by definition, violate transverality conditions because budget constraints are met temporarily but violated permanently when the irrationality/bubble/Ponzi is uncovered. This means that much theorizing by the New Keynesians and New Classicals should be rejected when there are real-world financial crises, such as the meltdown of 2008. An explanation is given by Buiter, who thinks that real-world markets cannot be modeled without violating transversality. Buiter in effect states that the new macro models assume honesty when there is none. But Buiter perhaps overstates his case, in that there is work on macro models with irrationality and contract enforcement problems (see here), or that Buiter is himself not that well informed (see Andolfatto on Buiter). Possibly, the new work is still a bit too new (there is an implied admission by the current theorizers that the mating of theory with reality is not quite there).