What does this diagram do for us?

Time for a big word: Naturality of the long exact sequence in homology and commutativty of the diagram shown means that not only will the map $f$ descend to the homology groups but it will descend in such a way that the diagram below will be commutative.

$$\xymatrix{ \cdots \ar[r] & H_n(A) \ar[r] \ar[d]_{f_\ast} & H_... ...{X}) \ar[r] & H_n(\overline{X}, \overline{A}) \ar[r] & \cdots }$$