Border, margin, verge, edge, rim, brim, brink mean the line or relatively narrow space which marks the limit or outermost bound of something.
A border is the part of a surface which is just within its boundary line or it may be the boundary line itself.
Margin denotes a border of definite width usually distinguished in some way from the remaining surface; it also applies to the space immediately contiguous to a body of water.
Verge applies to the line or to a very narrow space which sharply marks the limit or termination of a thing (as a surface or an expanse).
Verge may also be applied to the extreme limit of something with an implication that it is being approached either from within or from without.
An edge is a sharply defined terminating line made by the converging of two surfaces (as of a blade, a dish, a plank, or a box).
Edge often implies sharpness (as opposed to bluntness) and therefore power to cut.
It is this implication that comes out strongest in extended use where it often suggests asperity, trenchancy, or keenness.
Rim usually applies to the verge or edge of something circular or curving.
Brim applies to the inner side of the rim of a hollow vessel or to the topmost line of the basin of a river, lake, or other body of water.
Brink denotes the edge of something steep (as a precipice); thus, one would speak of the river’s brink when stressing the abruptness of the bank or shore but of the river’s brim when the notion of the close approach of the water to the basin’s rim is uppermost in mind.
Brink may also be used of immaterial things with the implication of a possibility or risk of abrupt transition (as front one state to another).