For f(S,t) with S following a diffusion, the differential picks up an additional half times the second derivative times variance term, beyond what ordinary calculus would give.
Where the extra term comes from. Brownian motion paths have non-zero quadratic variation - dW squared behaves like dt rather than vanishing - so second-order effects survive in the limit.
Why it matters. This term is exactly the gamma contribution, and applying Ito to a hedged option portfolio is how the Black-Scholes equation is derived.