Over dispersion in statistics means the presence of higher variability in a data set than can be obtained from a given simple statistical model. Simply put, it refers to a dataset with higher than the expected variance and finds wide applications in biological sciences. Over dispersion exists when the observed variance is higher than the variance of a theoretical model.
Conversely, under-dispersion means that there will be less variance in data that it is predicted. This is a common feature given that in practice populations are frequently heterogeneous leading to over dispersion. The mean reason behind over dispersion is that population is always much more heterogeneous than it is predicted to be. In Poisson distribution, over dispersion is often encountered and the variance cannot be adjusted independent of the mean. Hence, a negative binomial distribution can sometimes be used instead in which the mean itself is a random variable.
Over dispersion is observed in binomial distribution as well and to provide a better fit to the observed data, beta-binomial distribution or a Bernoulli distribution can be used that introduces a normal random variable into logistic model. The normal distribution has two parameters, the mean and the variance and hence any data with finite variance will not be over dispersed when modeled into the normal distribution.