A fibre is a projection. The simple case on the right is intended to show a projection of a three dimensional object onto a two dimensional surface. This is just a projection from a product space. However a projection along a fibre is more general than that, locally it still looks like a product space, but globally it can look very different. 

A standard example of this is a spiral, which can be projected onto a circle since any local section corresponds to a section of a circle. Locally a section of the spiral looks like a section of the circle, globaly the circle and the spiral are different.  
A similar example is the MÃ¶bius band being mapped onto a cylinder. 
Fibrations of Graphs
theory: topological graph theory
undirected graph:  covering projection 
directed graph:  fibration (weaker form of covering projection) 
Here is an example for directed multigraphs. Each node in the top graph maps to the bottom graph (fibration). The corresponding node always has the same number and colour of incoming arcs (but not necessarily outging arcs). This gives some sort of local invariance. 