The poisson distribution is specified by one parameter lambda that represents both the mean and variance of the distribution. Thus, it assumes that the mean and variance are roughly the same. If that is not the case, usually when the variance is larger than the mean, overdispersion occurs.
When overdispersion is present, using Poisson regression can produce unreliable conclusions, as the underlying distribution might not be Poisson. If the assumption of the mean and variance being equal is not met, it is common to model the data using the Negative Binomial distribution with a dispersion parameter, which can be thought of as analogous to the sigma parameter that measures the spread of a normal distribution.