Edge percolation (often called bond percolation) refers to edges being randomly removed from a graph (a lattice): typically, each edge has probability $p\in (0,1)$ of surviving, independent of other edges. In line with original meaning of _percolation_ (fluid transit), one may think of the removed edges as closed, i.e., made impermeable. How far the fluid may spread in the graph depends on the size of connected component. One often sees a _phase transition_ : emergence of a unique giant component when $p$ exceeds some value $p_0$.
Some references: Wikipedia page, Pierre Nolin's lecture slides.