What is a map of pairs?

First of all, a pair $(X,A)$ is simply a space $X$ and a subspace $A \subset X$. We then say $f$ maps a pair $(X,A)$ to a pair $(Y,B)$ if $f$ maps $X$ to $Y$ and $f(A) \subset B$.

If we have such a map then it will not only descend to a map on $H_n(X)$ but $H_n(A)$ and $H_n(X,A)$ as well.