The bootstrap approximates the sampling distribution of a statistic by repeatedly resampling with replacement from the observed data. If $\hat\theta=s(X_1,\dots,X_n)$, then the bootstrap distribution of $\hat\theta^*$ estimates its variability.

Bootstrap methods leverage the empirical distribution as a plug-in estimate of the unknown population distribution. They are especially useful when analytic standard errors are difficult to derive.

Monte Carlo methods approximate expectations and probabilities by simulation. For example, $\mathbb{E}[g(X)]\approx \frac{1}{M}\sum_{m=1}^M g(X^{(m)})$. This underlies Bayesian computation, uncertainty propagation, and randomized algorithms.

The jackknife, subsampling, wild bootstrap, block bootstrap, and parametric bootstrap address different settings such as heteroskedasticity, dependence, and time series.