From Wikipedia:Thus, "parts" or "cells" are subsets of the set that is being partitioned, K in this case. If a partition is generated by an equivalence relation, as in this case, partition "parts" are also called equivalence classes.A partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X.

To understand the relation R better, I suggest drawing a 10 x 10 grid on the coordinate plane and connecting points that are related by R. Note that if two points (x1,y1) and (x2,y2) are related, then they have the same sum of their coordinates. What does it say about their location?