  For $m\in \mathbb{N}, m\geq 1,$ we determine the irreducible components of
the $m-th$ jet scheme of a normal toric surface $S.$ We give formulas for the
number of these components and their dimensions. When $m$ varies, these
components give rise to projective systems, to which we associate a weighted
graph. We prove that the data of this graph is equivalent to the data of the
analytical type of $S.$ Besides, we classify these irreducible components by an
integer invariant that we call index of speciality. We prove that for $m$ large
enough, the set of components with index of speciality $1,$ is in 1-1
correspondance with the set of exceptional divisors that appear on the minimal
resolution of $S.$
