A fat tailed probability distribution is one in which extreme events are more probable.
For example the normal distribution is typically a very good fit over a wide range of the most likely outcomes in finance. However, it is generally accepted that extreme events such as crashes are more probable it suggests.
This means that a model based on the normal distribution will give good estimates over a reasonable range, but still under-estimate extremes. For example a model might accurately estimate the chance of a share price falling 10%, but greatly under-estimate the chance of the market falling 50%.
Although there is no great difficulty in producing fat tailed probability distributions, actually producing useful valuation or risk models with these is more difficult.
A great advantage of the normal distribution is that it is mathematically easy to handle. Assuming it makes it possible to derive formulae such as Black-Scholes. Using a fat-tailed distribution would make deriving a formula more difficult or impossible. It is sometimes possible to avoid the need for formulae by modelling using a computational approach such as a Monte-Carlo simulation.