Conjugacy classes Of elements of $SL\_{2}(\mathbb{Z})$ can be characterized by its trace. If $\mid T(m)\mid=1 $, then it is an element of finite order, if $\mid T(m)\mid =2$, then m is conjugate to a matrix up to sign,$ (1 n) (0 1)$ ( row notation) where $ n $ a natural number. in other case, there is an expansion in words $ML^{n\_{i}} MR^{n\_{i}}$, where ML is the matrix $(10)(11)$, and MR is the matrix $(01)(11)$ (Again, row notation). Is there a Classification or method for the determination of the centralizers of these elements, or the Normailizers of the subgroups generated by them ? 



Centralizers  of  elements  in $Sl_{2}(\mathbb{Z})$
This is a simple computation (write down your matrix, and see what it means for a two-by-two matrix to commute with it, or to normalize the subgroup). When doing the computation, it is useful to remember that two matrices commute if (and only if) they have the same eigenvectors.


**EDIT** In view of the fancy-pants arguments in the other answers, I do the computation for a parabolic in your notation $(1\ x)(0\ 1).$ Conjugating by $(a,\ b)(c,l\ d)$ gives you


$(1 - a c x,\ a^2 x)(-c^2 x, \ 1+ a c x).$
from the lower left corner we see that $c=0.$ That means (since we are in $SL(2, Z),$ that would not work in $SL(2, R)$) that either $a=d=1$ or $a=d=-1,$ so the normalizer in $PSL(2, Z)$ is just the subgroup itself, in $SL$ you throw in the center. The computation for elliptic (or hyperbolic) element is equally difficult.


