What Ito's lemma says and why quants need it

What is Ito's lemma, and why is it important in quantitative finance?

The chain rule for functions of a stochastic process. If $dX=\mu\,dt+\sigma\,dW$ and $f(t,X)$ is smooth, then $df=\big(f_t+\mu f_x+\tfrac12\sigma^2f_{xx}\big)dt+\sigma f_x\,dW$. The extra $\tfrac12\sigma^2f_{xx}$ term arises because $(dW)^2$ behaves like $dt$. It is how one finds the dynamics of an option price $V(S,t)$, leading to the Black-Scholes PDE once the $dW$ term is hedged away.