There are generally two definitions of a \emph{simplicial abelian group}, an abstract one and a very explicit one. We will start with the abstract one, and immediately show in pictures what the explicit definition looks like.
$$\delta_i: [n]\to[n+1], k \mapsto\begin{cases} k &\text{if } k < i;\\ k+1&\text{if } k \geq i. \end{cases}\hspace{0.5cm}0\leq i \leq n+1, \text{ and}$$
$$\sigma_i: [n+1]\to[n], k \mapsto\begin{cases} k &\text{if } k \leq i;\\ k-1&\text{if } k > i. \end{cases}\hspace{0.5cm}0\leq i \leq n$$
for each $n \in\N$. The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. So in a certain sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as follows.
\begin{figure}[h!]
\label{fig:delta_cat}
\includegraphics{delta_cat}
\caption{The category $\DELTA$ with the face and degeneracy maps.}
Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $F: \DELTA^{op}\to Ab$. And hence we can picture a simplicial abelian group as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $A$ is a contravariant functor.
Because a simplicial abelien group $A$ is a contravariant functor, these equations (which only consist of compositions and identities) also hold in its image. For example the first equation would look like: $ A(\delta_i)A(\delta_j)= A(\delta_{j-1})A(\delta_i)$ for $ i < j$ (again note that $A$ is contravariant, and hence composition is reversed). This can be used for a explicit definition of simplicial abelien groups. In this definition a simplicial abelian group $A$ consists of a family abelian groups $(A_n)_{n}$ together with face and degeneracy maps (which are grouphomomorphisms) such that the simplicial equations hold.
\subsection{Other simplicial objects}
Of course the abstract definition of simplicial abelian group can easilty be generalized to other categories. For example $\Set^{\DELTA^{op}}=\sSet$ is the category of simplicial sets.
\todo{sAb: as example do the free abelian group pointwise}