What kurtosis measures

What is kurtosis, and why do people in finance care about it?

The standardised fourth central moment, $\mathbb{E}[(X-\mu)^4]/\sigma^4$. It measures how much of the variance comes from extreme observations, i.e. how heavy the tails are. A normal distribution has kurtosis $3$, so excess kurtosis is kurtosis minus $3$; asset returns usually show positive excess kurtosis, meaning large moves are more frequent than a normal model predicts, which matters for VaR, option smiles and risk limits.