To welch basically means to trick or deceive someone by not making a payment on a bet. For example, 'we had a bet of five pounds that I would be able to lift the heavy table all by myself, I did, but that crook welched me". It refers to the dishonouring of a bet by not paying the rightful owner the predetermined amount which he deserves for winning the bet. It is a petty offence, though not regarded as a criminal one, as it is simply bound by an informal oral agreement. Hence, the person who has been cheated cannot do much but deal with it in a manner that he deems correct in a personal way. Anyway, such bets do not concern of large amounts of money.
Slang. Also means to evade the fulfilment of an obligation. If someone agrees to go and do something and then doesn't show up or changes the plan, he or she is said to have welshed on the arrangement or what was thought to have been an obligation. It matters to some degree how significant the obligation is perceived to be. If you agree to do something for me and I do not pay the agreed cost to you, it would be said that I welshed on you.