E[f(X)] >= f(E[X]) for convex f, with the inequality reversed for concave f.
Why it explains so much. Convexity is valuable precisely because of this - a convex payoff benefits from uncertainty. It is the formal reason a long-option position gains from volatility.
The concave side explains risk aversion: with concave utility, the expected utility of a gamble is less than the utility of its expected value, so a certain amount is preferred to a fair bet.
It is also why the mean of a lognormal exceeds its median.