  The basic properties of the Temperley-Lieb algebra $TL_n$ with parameter
$\beta = q + q^{-1}$, for $q$ any non-zero complex number, are reviewed in a
pedagogical way. The link and standard (cell) modules that appear in numerous
physical applications are defined and a natural bilinear form on the standard
modules is used to characterize their maximal submodules. When this bilinear
form has a non-trivial radical, some of the standard modules are reducible and
$TL_n$ is non-semisimple. This happens only when $q$ is a root of unity. Use of
restriction and induction allows for a finer description of the structure of
the standard modules. Finally, a particular central element $F_n$ of $TL_n$ is
studied; its action is shown to be non-diagonalisable on certain indecomposable
modules and this leads to a proof that the radicals of the standard modules are
irreducible. Moreover, the space of homomorphisms between standard modules is
completely determined. The principal indecomposable modules are then computed
concretely in terms of standard modules and their inductions. Examples are
provided throughout and the delicate case $\beta = 0$, that plays an important
role in physical models, is studied systematically.
