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Distribution-Free Methods
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Before computers, the mathematically "nice" features of the Gaussian, or normal, distribution meant that parametric methods were used most often. These days, the nice properties of the Gaussian no longer restrict our analyses because computers can brute force just about any solution.
Nonparametric tests
Many of the classical tests that require an assumption of normality have nonparametric counterparts where the actual observed values are replaced by their ranks. Nonparametric methods should be used when the data are known to not be normally distributed, for example, when the distribution of the data has a long tail or ordinal values (e.g., low, medium, or high) rather than continuous values (e.g., 0-100) with a unimodal distribution.
Resampling using a bootstrap or jackknife
Resampling with a bootstrap or jackknife algorithm creates multiple samples from a single sample. Typically bootstrap or jackknife samples are used to estimate variance when only a single sample is available, for example, to determine a confidence interval around an estimate. Resampling can also be used to test whether solutions from complex models are robust, that is, whether summary statistics are similar when the data change slightly.
Monte Carlo
Monte Carlo methods aren't necessarily distribution-free but are often lumped with these methods because they have a similar application and are also computer-intensive. For the bootstrap or jackknife, you resample from your original sample. In contrast, for Monte Carlo simulations, the data in the new samples are generated from an assumed model, not from your actual data. Thus, the data are simulated.