Consider the special case of the simplest complex of real vector spaces $\newcommand{\pa}{\partial}$
$$ 0\to U_0 \stackrel{\pa}{\to} U_1\to 0. $$
(Ultimately everything can be reduced to this simple situation via some algebraic tricks.)
This complex is acyclic iff $\pa$ is an isomorphism. By chossing bases in $U_0$ and $U_1$ appropriately we can represent $\pa$ as the identity matrix.
Assume this complex is acyclic and set $n=\dim U_0=\dim U_1$.
The torsion arises when $U_0$ and $U_1$ have additional data attached to them. Assume that $L_0$ and $L_1$ are lattices in $U_0$ and respectively $U_1$ such that $\pa(L_0)\subset L_1$. (These are finitely generated Abelian subgroups that span their correspondent ambient space. $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\bR}{\mathbb{R}}$ Think of the subgroup $\bZ^n$ of $\bR^n$.
A $\bZ$-basis $\newcommand{\ue}{\underline{\mathbf{e}}}$ $\newcommand{\uf}{\underline{\mathbf{f}}}$ $\ue_i$ of $L_i$ is also a $\bR$-basis of $U_i$. By choosing $\bZ$-bases $\ue_0$, $\ue_1$ of $L_0$ and respectively $L_1$ we can represent $\pa$ as an $n\times n$ matrix $M(\pa,\ue_0,ue_1)$.
Observe that if $uf_0$ and $\uf_1$ are other $\bZ$-bases of $L_0$ resp L_1$, then
$$\det M(\pa, \ue_0,\ue_1)=\det M(\pa, \uf_0,\uf_1)=|L_1/\pa L_0|, $$
where $|S|$ denotes the cardinality of the set $S$.
We see that if the associated complex is obtained from a complex of Abelian groups we can associated an invariant, the above determinant, or the order of the quotient $L_1/\pa L_0$. The torsion of this complex is the defined to be the number
$$\tau(\pa, L_0,L_1):=\frac{1}{|L_1/\pa L_0|}. $$
The chain complexes that appear in topology often come equipped with such lattices. Think of simplicial homology, or the chain complex associated to a $CW$-decomposition.
It turns out that for smooth manifolds the resulting invariant is independent of the triangulations used to define the torsion.
This is only the beginning of the story and I have omitted many important details. It should help you navigate the first part of my book on torsion where you will find many other descriptions and applications.