A properly honed edge will form an edge shaped like a perfect V.
Some combinations of steel and sharpening technique result in an edge shaped like a Y or y. That sliver of thin metal beyond the V shape is called a wire edge.
The wire edge is essentially a thin foil, useless for any typical cutting tasks since it has no strength. It will fold over, break off in sections or both, resulting in an instantly dull edge.
The wire edge can be difficult to grind away, since the thin foil tends to flop from side to side - bending away from the stone - rather than grind away cleanly.
One technique for removing the wire edge is to slightly increase the final bevel angle a few degrees, alternately honing both sides with low pressure until you cut the wire edge away from the V.
Difficult to describe without graphics - hope this helps!