Context: insurance, risk theory
Intuitive definition. An insurance company is ruined when it does not have enough cash reservers to meet claim payments. Generally considered as a Bad Thing. An recent example of this in Australia concerns HIH.
Mathematical definition. Consider a surplus process Ut, in which there is an initial reserve U0, a premium process and a claims process. The ruin event would be {Ut < 0}.
Because ruin is such a Bad Thing, a lot of mathematics (consuming countless hours of academic research) is involved in predicting such things happening, and ensuring that the insurance company has enough reserves so that the probability of ruin is acceptably small.