I was wondering if someone could one tell me where to locate this constraint because it is not a part of my relations menu. I would also like to have a better understanding of it.
An "On Edge" relation appears when using the "Convert Entities" sketch feature. This basically creates a reference to some edge in your model (or perhaps another model, like within an assembly).
It can cause issues if that geometry is removed or changed, which will very often break your On Edge relation.
So I am currently editing a part that has multiple variations but only uses 1 sketch. Is there a way that I can make so when I change the sketch the on edge constraint will mimic what has happened to the new sketch.
Depends on how you change it, I would say. For instance, if you just modify a few dimensions, then yes an on edge constraint should follow your changes. However, if you create or remove geometry from your sketch, an on edge constraint will not be able to perfectly follow that.
Basically this is the link between two different parts in assembly. You can delete constraint, break, lock and unlock references and redefine the undefined sketch different way.
2016 SOLIDWORKS Help - External References
Retrieving data ...