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][1] where you will find many other descriptions and applications.


  [1]: https://www3.nd.edu/~lnicolae/Torsion.pdf