An energetic community member has proposed a substantial tag wiki for the nonparametric tag. Because it includes so much, I thought it would benefit from the community participating in improving it here, rather than attempting to do so through the (extremely limited) wiki editing interface. Then, once the language stabilizes, we can modify the wiki itself. I am reluctant to approve it outright because, IMHO, not all the characterizations are as clear or correct as they ought to be for something intended as a general reference. But it's such a good start and such a devoted effort that I think it deserves our attention.
Suggestions can be made in the usual ways: replies, comments, and direct editing of this question itself. (Direct editing might be cleanest, but I'm not sure everyone will be able to carry that out.)
'Nonparametric' is a fairly broad term for a wide range of statistical analysis procedures that will still yield accurate results when classical assumptions are violated. As the term implies, the original idea was to develop techniques that did not attempt to estimate properties of the parameters (e.g., confidence intervals of the means) of the populations from which the data were sampled. Instead, these techniques were often based on ranks of the data and thereby afforded the calculation of correct p-values. A prototypical example would be determining the central tendency of a highly skewed distribution--the assumption of normality is violated, but methods based on the median would be unaffected.
Today the term is sometimes used to cover approaches (e.g., bootstrapping) that work in very different ways and are often used to work with estimated population parameters. As such, the concept is somewhat fuzzy. Nonparametric techniques can be compared and contrasted with several other types of statistical procedures, including semi-parametric & generalized additive models (which combine elements of both parametric and nonparametric approaches) and robust statistics that use alternative loss functions to minimize the influence of outliers and excessive skewness.
'Assumption-free' and 'distribution-free' are sometimes used as synonyms. Note that these terms, taken literally, are inaccurate descriptors of nonparametric procedures: Nonparametric analyses do entail assumptions, just fewer and ones that are more likely to hold. Moreover, nonparametric techniques typically rely on the uniform distribution (e.g., for rank-based procedures) or on estimated distributions (e.g., for bootstrapping), just not the normal distribution.
Lastly, although nonparametric usually connotes alternative statistical tests, it can be used to indicate an alternative approach to modeling. For instance, the simplest model of a dataset (assuming normality) is simply to estimate the mean and standard deviation, but a model of the data could be generated by kernel density estimation without ever calculating a mean.