If we denote L as the class of likelihood functions p(x|y), a class P of priors distributions p(y) is a conjugate family for L if the resulting posterior distribution p(y|x) is also in P for any likelihood function p(x|y). For example the Beta distribution family is a conjugate family for the Binomial. Meaning if you choose a Beta prior and your likelihood function happened to be a Binomial, you will have a posterior distribution which is also in the Beta family. This is quite useful because in this framework updating the prior amounts to just updating its parameter, which makes computation very easy.
However one should be careful since one might be tempted to choose a conjugate prior for the sole purpose of making computation easy although its does not correctly reflect prior knowledge about the variable. It is certainly not of any use to have a computationally convenient, but wrong model.
