A good criterion for the irreducibility of a fiber product is when one of the morphisms, say $f$, is flat with irreducible generic fibers. Flatness implies that $X\times_YZ\to Z$ is also flat. Hence every irreducible component maps dominantly to $Z$. The second condition implies that there is only one component of that type.

A good criterion for the irreducibility of a fiber product is when one of the morphisms, say $f$, is flat with irreducible fibers. Flatness implies that $X\times_YZ\to Z$ is also flat. Hence every irreducible component maps dominantly to $Z$. The second condition implies that there is only one component of that type.