Binomial family in logistic regression
From wikipedia:


  ..., the binomial distribution with parameters n and $\rho$ is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own boolean-valued outcome: a random variable containing a single bit of information: success/yes/true/one (with probability $\rho$) or failure/no/false/zero (with probability $\rho = 1 − \rho$).


So if you know that logistic regression is performed in order to model a binary output variable to some modelling question (i.e. to give 0 or 1, yes or no, etc.), it would make sense to base any probabilistic assumptions on a distribution, which shares this feature. Therefore, a binomial distribution may make sense compared to a continuous distribution, such as a Gaussian or Cauchy.