Broadly speaking, there are two kinds of statistical analysis that can be used to analyse data. One kind of analysis is the class of parametric methods where we use a model that assumes some particular group of possible distributions (e.g., a normal distribution, an exponential distribution, etc.). The other kind of analysis is the class of non-parametric methods where we use a model that makes minimal assumptions about the distribution of the data, and includes all common families of distributions, but also pretty much any distribution that is not a common form.
Parametric methods generally involve a model-fitting process based on the mathematical structure of the assumed form of the underlying distribution; once we fit the model we use diagnostic tests to check if the assumed model form is a reasonable representation of the data. These models usually allow the data to be represented by a fixed number of parameters (hence the name 'parametric') and the complexity of the model does not increase as we get more data. OLS linear regression analysis and standard generalised linear models are done in this way. Non-parametric methods generally involve a model-fitting process based on a very broad structure that allows the complexity of the representation of the underlying distribution to increase as we get more data. These methods allow for almost any underlying distributional form (counting out some pathological cases) and they usually model the data using broad representations that add complexity as we get more data. Spline and kernel-based methods are done in this way.
The field of non-parametric statistical methods is absolutely enormous; there are shelves of books on this topic in most university libraries. For that reason, it is not possible to give an account of the field in a stackexchange.CV answer. However, if you would like to learn more about this field, a useful starting point would be to look at a primer on non-parametric density estimation.