With the jargon-free version of this concept I can think of an example. To recompute the sum of the elements of a vector when one of the elements change you just add the new - old value to the sum. I'm I right?

Yes; Or: If you have a very long list of integers, for example [1, 2, ..., 12351234] and your output would be the same list but all numbers increased by one (keeping the index/order): [2, 3, ..., 1235123]. If you now add a new element to the end of the initial list, you'd only have to add one element to your output instead of iterating over the whole list.