Relaxation is the process of updating a distance estimate when you find a better path. If you have dist[v] and discover a path through with length dist[u] + w, you relax the edge by setting dist[v] = min(dist[v], dist[u] + w).
This term comes from the idea of tightening or loosening constraints. You are finding the tightest (shortest) distance estimate. Relaxation appears in Dijkstra, Bellman-Ford, and DAG shortest path algorithms.