According to the theoretical distribution, events that deviate from the mean by five or more standard deviations ("5-sigma event") are extremely rare, with 10- or more sigma being practically impossible. However, under many applications, such events are more common than expected; 15- or more sigma events have happened in finance, for example. Because the real-world commonality of high-sigma events is much greater than in theory, the distribution is "fatter" at the extremes ("tails") than a truly normal one.
In finance, fat tails are considered undesirable because of the additional risk they introduce. For example, an investment strategy may have an expected return, after one year, that is five times its standard deviation. Assuming a normal distribution, the likelihood of its failure (negative return) is less than one in a million; in practice, it may be higher. Normal distributions that emerge in finance generally do so because the factors influencing an asset's value or price are mathematically "well-behaved", and the central limit theorem provides for such a distribution. However, traumatic "real-world" events (such as an oil shock, a large corporate bankruptcy, or an abrupt change in a political situation) are usually not mathematically well-behaved.
Fat tails, or abnormal sigma events are the prime reason why naked short option writers using mathematical probability models for their trading very often end up doing their arse by unexpected mathematical improbabilities in price action.