Prominence is how far a peak sticks up. In other words, how separate a given peak is from nearby higher peaks. So you can tell the difference between a truly separate peak versus a bump on the ridge leading up to a higher peak. More on this later.
To calculate prominence of any peak, you look at the topo maps to see how far you must drop before you start going up a higher peak. Note the words "must drop". That tells you that when calculating prominence, you are looking for the highest ridge connecting to a higher peak. There may be other higher peaks nearby, but the drop is on the highest route, not across a valley. For example, if you are on the summit of Mount Tiedemann, you must drop 848 meters to get to a higher mountain (which in this case is Mount Waddington. The low point is called the "Key Saddle" of Tiedemann. Think of going along a ridge connecting the peak with higher ground. The key saddle is the lowest point on that ridge. Another way to think of prominence is to envision flooding the world till the subject peak is the highest point on its own island. That island will have exactly one point where it attaches to an adjacent island with a higher peak. That is the key saddle of the lower peak. The technical definition of "Mountain Prominence" is the vertical distance between a peak and the lowest contour line surrounding that peak and no higher peak.
What use is the concept of prominence? It is a statistic that tells you how separate the mountain is from its "parent" peak. So when you are talking about a mountain, you typically quote its height and its prominence. For example, compare the following mountains:
Height Prominence -------------------------------------------
Bodyguard Peak 3325m 55m
Good Hope 3242m 1497m
From the above you can see that Good Hope is much more of separate mountain than Bodyguard Peak, even though Bodyguard is higher. Bodyguard is just a bump on the ridge leading up to Mount Waddington.
So now you know why we bother to measure quote prominence statistics - otherwise, if you look only at height, you can't tell how separate the mountain really is. Prominence is NOT a measure of how hard the peak is to climb. A low prominence peak might be a quite difficult climb and take all day. Just because prominence is only 40m doesn't necessarily mean you could stroll down the ridge from the higher peak. Prominence is strictly a numerical calculation from the topo maps. Everyone would get the same answer. It is not a subjective number based on any other factors.
Prominence is easy to understand when the two mountains are side by side, but the definition applies to all mountains, even when the parent peak is quite distant. The highest mountain in an area may have its prominence determined by a pass that is quite distant from the peak. For example, The prominence of Mount Robson is determined by Yellowhead pass, and to get to something higher on the other side of the pass, you have to go over a thousand km south, towards Gannett Peak. The prominence of Mount Robson is 2824m above Yellowhead Pass, which separates it from the higher Gannett Peak."
Note that prominence is NOT just the drop from the steepest face of the mountain to the valley below.
How do you find the correct "key saddle"? What if there are multiple possible higher peaks? The key saddle is the route on the HIGHEST route to higher ground.