A famous example of a topologically nontrivial bundle is the Moebius strip which is a nontrivial bundle over the circle. A topological trivial analogue would be a cylinder.
Is it possible to have a nonzero curvature on a topologically trivial fiber bundle? (And if yes, is there any way to visualize this?)
Let $P\rightarrow M$ be a non flat bundle, let $U$ be a contractible open subset of $M$, the restriction of $P$ to $U$ is trivial, but the curvature is not always zero. To have a visual picture of this, note that the curvature defines the Lie algebra of the holonomy of the connection (Ambrose Singer) around small loops that you can be supposed contained in $U$.
A concrete example will be the tangent bundle of the sphere and its associated frame bundle, that you restrict to an hemisphere.