The "default", the most commonly used and described, distribution of choice for count data is the Poisson distribution. Most often it is illustrated using example of its first practical usage:
A practical application of this distribution was made by Ladislaus
Bortkiewicz in 1898 when he was given the task of investigating the
number of soldiers in the Prussian army killed accidentally by horse
kicks; this experiment introduced the Poisson distribution to the
field of reliability engineering.
Poisson distribution is parametrized by rate $\lambda$ per fixed time interval ($\lambda$ is also it's mean and variance). In case of regression, we can use Poisson distribution in generalized linear model with log-linear link function
E(Y|X,\beta) = \lambda = \exp\left( \beta_0 + \beta_1 X_1 + \dots + \beta_k X_k \right)
that is called Poisson regression, since we can assume that $\lambda$ is a rate of Poisson distribution. Notice however that for log-linear regression you do not have to make such assumption and simply use GLM with log link with non-count data. When interpreting the parameters you need to remember that, because of using log transform, changes in independent variable result in multiplicative changes in the predicted counts.
The problem with using Poisson distribution for the real-life data is that it assumes mean to be equal to the variance. Violation of this assumption is called overdispersion. In such cases you can always use quasi-Poisson model, non-Poisson log-linear model (for large counts Poisson can be approximated by normal distribution), negative binomial regression (closely related to Poisson; see Berk and MacDonald, 2008), or other models, as described by Stephan Kolassa.
For some friendly introduction to Poisson regression you can also check papers by Lavery (2010), or Coxe, West and Aiken (2009).
Lavery, R. (2010). An Animated Guide: An Introduction To Poisson Regression. NESUG paper, sa04.
Coxe, S., West, S.G., & Aiken, L.S. (2009). The analysis of count data: A gentle introduction to Poisson regression and its alternatives. Journal of personality assessment, 91(2), 121-136.
Berk, R., & MacDonald, J. M. (2008). Overdispersion and Poisson regression. Journal of Quantitative Criminology, 24(3), 269-284.