In order to give a reasonable definition of this term, it helps to be reminded of some basic facts regarding mountains.
With all that, we can define prominence:
The prominence of a mountain is the elevation differential between the mountain’s summit elevation and its highest pass/saddle connecting it to its parent.
The exception is for the land mass’ highest mountain. In this case its prominence is its elevation, relative to sea level.
Suppose Mountain A has a summit elevation of 10,000 feet. Its parent is Mountain B and has an elevation of 10,400 feet. The highest saddle connecting the two is 7,500 feet – in other words, if you were to walk from Mountain A’s summit to Mountain B’s summit, you would be forced to descend to 7,500 feet. Hence, Mountain A’s prominence is 2,500 feet. Note: Mountain B’s prominence is not defined by this same saddle! It’s defined by the same definition – requiring one to travel to Mountain B’s parent’s via its highest saddle.
When mountains are grouped in one physically-connected range, it’s fairly easy to determine the peaks’ prominences. The tricky part is when we start looking at supersets of ranges – tracking lineages from one range to another, forcing one to descend onto the valley floors where the ‘ridges’ are very indistinct and must be found by computer analyses.
The whole point of prominence is to give an objective sense of the ‘size’ of a mountain. Mountains with high prominence ‘stand out’ against the countryside. A peak with an elevation of just 4,000 feet, but with 3,000 feet of prominence, is a very ‘large’ peak and stands out as a very visible peak. On the other hand, a peak with an elevation of 12,000 feet might only rise 1,000 feet above its neighbors and as a result, does not have much prominence and is not as visually distinct.
Prominence does not necessarily equate to the actual gain needed to hike the peak (although the correlation is strong). A highly-prominent peak might have a road most of the way up, leaving just a relatively small hiking gain to the top. The converse is also true: a low prominent peak might have difficult road access and force its climbers to start low and gain more in elevation.
By and large prominence offers a useful metric for a mountain’s measure. Here are some examples:
The most prominent mountain in the eastern United States is Mount Mitchell in North Carolina, with a huge 6,000+ feet of prominence. Its lineage actually takes it across the country into the Rocky Mountains!
Similarly, the highest mountain in North America is Denali in Alaska at 20,320 feet. But Aconcagua in South America is the American land mass’ highest point, hence Denali’s ‘parent’ is Aconcagua (Aconcagua has no parent). Where is the ‘highest saddle’ connecting the two? It’s apparently somewhere in Nicaragua at about 150 feet elevation.
The Kaibab Plateau in northern Arizona is a vast plateau that rises over 3,000 feet above all its surrounding plains and saddles. Yet once on top the plateau, all one sees is gentle hills. And yet it highpoint is one of the most highly prominent places in Arizona!
Fortunately, the notion of prominence seems to merge well with most subjective measures of a mountain. For a much further detailed study of this concept, I suggest you contact Adam Helman and get his take on it. He sells a very detailed book on the subject, chock full of lists.