nLab
fiber

Given a bundle $p:E\to B$ and a global element $x:1\to B$, the fibre or fiber $E_x$ of $(E,p)$ over $x$ is simply the pullback

$$\begin{array}{ccc}{E}_x& \to & E\\ \downarrow & & {\downarrow }_p\\ 1& \stackrel{x}{\to }& B\end{array}$$\array { E_x & \to & E \\ \downarrow & & \downarrow_p \\ 1 & \stackrel{x}\to & B }

if it exists.

In a fiber bundle, all fibres are isomorphic to some standard fibre $F$ in a coherent way.