I know that a von Neumann algebra on a separable Hilbert space can be (uniquely) decomposed into factors. But is there any non-trivial example showing how we can explicitly compute it?

Non-trivial: I mean it is not a factor itself, not abelian or of finite dimension.

typehas a specific meaning in the factor theory: there are type ${\rm I}_n$, ${\rm I}_{\infty}$, ${\rm II}_1$, ${\rm II}_{\infty}$, ${\rm III}_{0}$, ${\rm III}_{\lambda}$, ${\rm III}_{\infty}$. The type does not characterize completely the factor, except in the amenable case (${\rm III}_{0}$ excepted). $\endgroup$ – Sebastien Palcoux Aug 24 '17 at 6:24