Quicksort is a divide-and-conquer algorithm that repeatedly "sorts" pivot values on consecutively smaller subarrays until every value has been a pivot so that every value becomes sorted. The "sorting" of a pivot is known as partitioning.
Partitioning an array is the act of moving data around a chosen Pivot. Specifically, it rearranges data so that Left <= Pivot <= Right. The result is that the Pivot is "sorted" with respect to Left and Right, but the Left and Right sides are not themselves sorted. This is then repeated on the Left and Right subarrays. The process of recursive partitioning ultimately results in a fully sorted array.
Merge sort is a divide-and-conquer algorithm that divides a list by half into left and right subarrays until the size of the subarrays are length 1. It then pair-wise merges left and right subarrays to form a sorted result. The merges continue until the entire set is rebuilt into a single sorted array.