Let $X$ be a complex manifold and $Y\subseteq X$ a submanifold. It is well known that the Thom class of the normal bundle of $Y$ over $X$ is the Poincaré dual [Y]. I read that this result is important because we can give an explicit construction of the Thom class, specially when the bundle is trivial.
Is it true that the normal bundle in this case is always trivial? Where can I find the explicit constructions of the Thom class (at least for trivial bundles)? What extra information gives us the thom class about the Poincaré dual?