(see also Chern-Weil theory, parameterized homotopy theory)
The trivial fiber bundle over $B$ with fibre $F$ is simply the Cartesian product $B \times F$, equipped with the projection map to $B$ to make it a fiber bundle over $B$
A fiber bundle which not the trivial bundle but is isomorphic to the trivial bundle is called trivializable and the chosen isomorphism is called the corresponding trivialization.
If $F$ is a vector space, then the trivial $F$-fiber bundle is a trivial vector bundle, etc.
If a field bundle is trivial with fiber $F$, then the corresponding field theory is called a sigma model with target space $F$.