Colloquial verb used (at least in the U.S.) to refer to the act of failing to pay one's half of a bargan. Most often used when referring to a bet or a deal with money involved.
I was looking forward to selling my Everquest character to this guy on eBay, but the bastard welched on the deal.
From what I have seen, "welch" is most frequently used as an intransitive verb, with the connector "on the" used with the object. The subject of the clause is always the one failing to pay, such as "the bastard" in the above sentence.
The OED defines this term (under welsh) as a transitive verb as well, going all the way back to 1857. It is "of obscure origin," presumably unrelated to the nationality of Wales. (Contrast with "to Jew down" for "negotiate," which surprised me a bit when I first heard it.)