Given an array A of n elements, prefix sums are a very powerful technique that allows you to calculate the sum of any range of elements [l, r] for 1 <= l <= r <= n in O(1) constant time with the help of O(n) pre-processing.
The trick is to pre-calculate in another array prefix, of cumulative sums, such that for each i from 1 to n, prefix[i] stores the sum a[1] + a[2] + ... + a[i]. Then, the sum of elements in the range [l,r] can be easily found out as prefix[r] - prefix[l-1].
Peltorator on Prefix Sums: Great educational video, however it is in Russian with english subtitles.
Here are some links to practice prefix sum based problems:
Challenge Problems: