Quicksort is a sorting algorithm that works by selecting a pivot element and partitioning the array around it such that elements smaller than the pivot come before and elements larger than the pivot come after. It then recursively applies this process to the subarrays until the entire list is sorted. The algorithm demonstrates this process step-by-step on an example array, partitioning the list around the pivot elements and recursively sorting the sublists until the array is fully ordered.