The map p:E \rightarrow B is a fiber bundle with fiber F if for every point x \in B there is an open neighborhood U_{x} \subset B and a "fiber preserving homeomorphism" \phi_{U_{x}} : p^{-1}(U_{x}) \rightarrow U_{x}\times F.

In particular, the projection map X \times F \rightarrow X is the trivial fibration over X with fiber F.

What I don't understand is the following example.

Let S^{1} be the unit circle with basepoint 1 \in S^{1}. Consider the map f_{n} : S^{1} \rightarrow S^{1} given by f_{n}(z)=z^{n}. Then f_{n} : S^{1} \rightarrow S^{1} is a locally trivial fibration with fiber a set of n distinct points (the nth root of unity in S^{1}

What is the U_{x} in this example? is it an arc? Why is the fiber F the nth root of unity here? What is this "fiber"? Is there a geometrical meaning to the U_{x} \times F here?

I do not really have any background on topology. So appreciate if someone can explain this to me. Thanks!