A "butterfly" nose is when a dog has patches of bright pink on its nose leather. These patches are randomly located and can cover any amount of the nose, from just a tiny pink spot to almost the whole nose. Butterfly noses sometimes occur on dogs with the extreme white spotting pattern ( e.g. white Boxers, Bull Terriers and Dogo Argentinos), but generally they're associated with merles. Not all merles have butterfly noses though - double merles and normal merles with less dark patches than normal are more prone to them, and they are very common in harlequin Great Danes.
Butterfly noses are created when parts of the nose have no pigment (the colour pink is associated with a lack of pigment). The merle gene dilutes random parts of the pigment on the coat and nose, creating grey areas on the coat and pink areas on the nose. Harlequin Great Danes have an extra gene which dilutes the grey areas on the merle to white, so this extra strong dilution means harlequins are highly likely to have a butterfly nose.
The non-pink parts of the nose can be liver, blue or isabella if the dog is bb (liver) and/or dd (dilute). Livers and isabellas often have very light noses anyway, sometimes bright pink, so a butterfly nose may not show up on a liver or isabella merle (the whole nose may just appear pink). The first example below shows a liver merle with a visible butterfly nose