See here for definition of relative homology group. Making precise the idea of ‘homology of X modulo A.’ – video

Definition of relative homology, denoted $H_n(X,A)$ See also here. See here. Defined on relative chain groups, for which we need to give the subset $A$ a Singular complex (or other complex) structure, and maybe change $X$'s structure to make $A$'s complex a subcomplex, and the Chain group of $A$ be a Subgroup of the chain group of $X$, so that we can define their quotient. Note that different complexes on the same space give the same homology, so this arbitrary choice doesn't affect the answer, and relative homology is thus well-defined.

Homology long exact sequence

Long exact sequence relates relative homology to absolute homologies

Main way to compute relative homologies.

Long Exact Sequence of Pair. See more at Exact sequence, and Homology long exact sequence

Some applications of relative homology

Some examples:

$H_i(D^n, \partial D^n) = \begin{cases}\mathbb{Z}~~\text{for}~i=n\\ 0~~ \text{otherwise}\end{cases}$

$H_n(X,x_0) \cong \tilde{H}_n(X)$ for all $n$ since $\tilde{H}_n(x_0) = 0$ for all $n$.

Similarly to normal homology, homotopic maps between pairs of spaces induce the same homomorphisms between the correponding relative homologies.

Excision theorem

A useful tool for finding relative homologies too, by removing some part of the space..

Proposition 2.22 (book)