contrast is the "difference" between the brightes and darkest parts of a print. You may apply this to the overall print, to a part of a print or even to tiny deatails. This is not to be meant "densitometrically", i.e. if there is a tiny spot with Dmin and another tiny spot with Dmax on "landscape in fog", the overall contrast is still low.
Local Contrast is often associated with "mid tone separation", i.e. how simple things, sturctures, textures etc. can be differentiated.
Detail Contrast on the other hand, is directly associated with "sharpness", i.e. how clear the eye can seperate tiny stuctures. This is why higher paper grades usually look sharper.