The topic you're looking for is 'density estimation'.
The first thing to note is that you will probably not find that a mathematically familiar distribution fits your data (unless you work on radioactive decay, when the Poisson really is what you want...)
The second thing to note is that one may, subject to various theorem-specific technical conditions, approximate any distribution arbitrarily well given enough data with a mixture of more familiar distributions e.g. from the exponential family.
For example, a simple approach to estimating the distribution of continuous data might be a finite mixture of Gaussians. Maximum Likelihood estimates for parameters are available using an EM algorithm. (EM algorithms are easiest to implement for exponential family). Naturally other methods are possible. McLachlan et al. 2000 is a good place to start reading. The R package flexmix will help you fit these models.
The basic issues that arise in this exercise involve not having enough samples in all parts of the data space - the curse of dimensionality - and consequently the question of how to trade-off bias and variance in one's density model.
Density estimation is in this sense inherently harder than building conditional models such as regression and classification. For example, in the latter problem one needs only to characterise the area of the data distributon where the class decision boundaries lie rather than all of it.
Finally, note that for many problems the distribution of your data is not relevant. A standard motivation for regression modeling is to observe that some P(X, Y) can be decomposed into P(X) the density of the input data and P(Y | X) the conditional density of the target data. If it is further assumed that the parameters governing these two distributions are a priori independent (as they will be when distinct mechanisms assign X and generate Y from it) then the density P(X) can be ignored when trying to learn about P(Y | X), which is convenient because the latter is typically a much lower dimensional distribution to characterise.