Given a function $f\colon X \longrightarrow Y$ a fibre is an inverse image of an element of $Y$ That is given $y \in Y$ $f^{-1}(\{y\}) = \{ x \in X \mid f(x) = y \}$ is a fibre.

Example: Define $f\colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ by $f(x,y) = x^2 + y^2$ Then the fibres of $f$ consist of concentric circles about the origin, the origin itself, and empty sets depending on whether we look at the inverse image of a positive number, zero, or a negative number respectively.

Example: Suppose $M$ is a manifold, and $\pi\colon TM\to M$ is the canonical projection from the tangent bundle $TM$ to $M$ Then fibres of $\pi$ are the tangent spaces $T_x(M)$ for $x\in M$