Exactness Axiom of Homology Theory Axioms we are using for Homology Theory:
1) Homotopy:  if $f$ and $g$ are homotopic, then $h_{n}(f) = h_{n}(g)$
2) exactness:  each map $f:(X,A)\to (Y,B)$ gives us a commuting ladder of long exact sequences (the top bar of which I have included below in my question)
3) Excision:  if $(X,A)$ is a pair and $C\subset A$ with the closure of $C$ contained in the interior of $A$, then the inclusion $e:(X-C,A-C)\to (X,A)$ induces an isomorphism $h_{n}(e):h_{n}(X,A)\to (Y,B)$.
Exactness axiom:
For each $f:(X,A)\to (Y,B)$ there is a commuting ladder of long exact sequences:
$\dots \to h_{n}(A,\phi)\to h_{n}(X,\phi)\to h_{n}(X,A)\to h_{n-1}A\to ...$
My question:
Based on my notes, I can't find a definition for $h_{n-1}A$, nor the map $h_{n}(X,A)\to h_{n-1}A$ (which is, however, labelled as $\partial_{(X,A)}$)
I browsed some online sources and found that it is refered to as a boundary map.
But what is its definition?   (same question for the space $h_{n-1}A$).
Thanks so much in advance!
 A: Let me try to answer the question "what is the boundary map $\partial: H_n(X, A) \to H_{n-1}(A)$?"
As noted in the comments, this is part of the axioms of a homology theory, so actually the official answer is that it's given to you as part of the homology theory. Obviously, this is not very satisfying, so instead let me answer the question "what is the boundary map in singular homology?"
The answer is that if your given homology theory is constructed as the homology of some chain complexes (which singular homology certainly is!), the boundary map is something you get for free for abstract nonsense reasons (i.e. diagram chasing).
Given a pair $(X, A)$, for each $k$ we have the following commuting diagram
$$
\begin{array}{ccccccccc}
0 &\to& C_k(A) &\xrightarrow{i_k}& C_k(X) &\xrightarrow{q_k}& C_k(X)/C_k(A) &\to& 0 \\
&  & \downarrow & & \downarrow & & \downarrow & &  \\
0 &\to& C_{k-1}(A) &\xrightarrow{i_{k-1}}& C_{k-1}(X) &\xrightarrow{q_{k-1}}& C_{k-1}(X)/C_{k-1}(A) & \to & 0
\end{array}
$$
where $i_k$ and $q_k$ denote inclusion and quotient, respectively, and all the downward arrows are $\partial_k$. Suppose I have some element of $H_k(X, A)$. Then I can pick some relative cycle $\tilde{c} \in C_k(X)/C_k(A)$ representing it, and since $q_k$ is surjective I can pick some $c \in C_k(X)$ such that $q_k(c) = \tilde{c}$. Consider $\partial_k c \in C_{k-1}(X)$. By commutativity, of the diagram, $q_{k-1} \partial_k(c) = \partial_k q_k(c) = \partial_k \tilde{c} = 0$, so $\partial_k c \in \ker q_{k-1}$. By exactness of the bottom row, there is some $b_{k-1} \in C_{k-1}(A)$ such that $i_{k-1}b_{k-1} = \partial_k c_k$. 
So let us define a map $H_k(X,A)$ by sending $[\tilde{c}]$ to $[b_{k-1}] \in H_{k-1}(A)$. A similar diagram chase shows that this map is well-defined (i.e. independent of all the choices we made). This is the well-known connecting homomorphism, which in homology we typically denote by $\partial_k: H_k(X,A) \to H_{k-1}(A)$. This construction is usually called the snake lemma.
