Regression - random error term
$\varepsilon$ is referred as a noise term with 0 mean. The distribution is random in the real world, but you can make assumptions on its distribution.

For example the Gaussian Process machine learning suggests that it follows a Gaussian distribution i.e :  $\varepsilon \sim \mathcal{N}\left(0, \sigma^ 2 \right)$.

The variance $\sigma$ of the latter distribution can be seen as an hyperparameter that we obtaining by maximizing the likelihood function, or by having prior information on the data set. 
You can find more information in this book Rasmussen, C. E., and C. K. Williams. "I (2006) Gaussian Processes for Machine Learning." (2006)

In some cases by looking at your a priori data, and estimating the possible error sources, you can a priori expect the type of the error distribution and try to assess this assumption a posteriori (mainly using data-driven method).