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

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

What I don't understand is the following example.

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

What is the $\displaystyle U_{x}$ 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 $\displaystyle U_{x} \times F$ here?

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