Error falls like 1/sqrt(N) - four times the samples for half the error.
The property that makes it indispensable: that rate is independent of dimension. Grid methods degrade exponentially with dimensions; Monte Carlo does not, which is why it wins for high-dimensional pricing despite being slow in one dimension.
Variance reduction beats brute force. Antithetic variates exploit symmetry, control variates subtract a correlated quantity with a known value, and importance sampling concentrates draws where the payoff is non-zero - essential for deep out-of-the-money options where almost all paths contribute nothing.