Consider inferring a distribution for a parameter, $\theta$ given data $x$. From Bayes Theorem we have that:
$$p(\theta|x) = \frac{p(x|\theta)p(\theta)}{p(x)} \propto p(x|\theta)p(\theta)$$
If $p(\theta)$ is a conjugate prior for $p(x|\theta)$, then the posterior, $p(\theta|x)$ will be of the same family as the prior $p(\theta)$.