An ink with high opacity will completely block anything underneath from showing through while one with low opacity will look more like a color wash over whatever is beneath. Encyclo defines opacity as the "property of an ink that prevents the substrate from showing through."
Inks of varying levels of opacity are sold on the market. Opacity is achieved through a careful balance of pigment particles and binding medium. This balance is discussed in depth by the China Coating Association.
Opacity of ink can be tested by dipping a flat knife in a sample of the ink and then spreading it across a black line printed on a piece of white paper. After it dries, the sample created is compared to a standard printed by the ink industry.
There are many uses for high opacity ink. Printing on black paper or dark-colored materials is possible with high opacity ink, even allowing white ink to show up clearly on black paper. Transparencies require high opacity to block light.
It may be more difficult to come up with uses for low opacity inks, but these are probably used at least as often. Low opacity inks enable the blending of colors necessary to create intricate designs and photographic images.