TITLE: The number of conjugacy classes of the simple group PSL(2,q)
QUESTION [7 upvotes]: If $q=p^a$  , where $p$  is a prime number, then I would like to know the number of conjugacy classes related to elements of order $p$  and $2$  in the simple group $PSL(2,q)$ .

REPLY [2 votes]: To amplify my comments (in community-wiki style:
1) Note that it's OK when discussing just unipotent elements (here those of order $p$) to work instead with the matrix group $G:=\mathrm{SL}(2,q)$, since $\mathrm{PSL}(2,q)$ is just the quotient by the center $\{\pm I\}$ which consists of semisimple elements (and is trivial for $p=2$).  Also, the methods are basically the same for all $q$ with $p$ fixed. 
2) Usually your question is part of a more general computation of conjugacy classes : size of each and total number of classes.   This may be easier to organize, since the total number of elements i $|G|=q(q-1)(q+1)$.   In any case, the tally of classes is most often part of the search for ordinary characters.   (For a short exposition, see my old paper in Amer. Math. Monthly 82 (1975), 21--39.)  Though I don't have most textbooks at hand, I'd also suggest looking at the arguments on page 230 of L. Dornhoff Group Representation Theory, Part A, Dekker, 1971.   This is now out-of-print but has useful short chapters on many topics including this standard example.    
3) Maybe it's worth emphasizing that the difference between $p=2$ and other primes for this purpose is that all nonzero elements of $\mathbb{F}_q$  are squares in the first case but only half the elements in the second case.   Thus a unipotent matrix in $G$ is conjugate for odd $p$ to just one of the matrices $$\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \quad \begin{pmatrix} 1 & \nu \\ 0 & 1 \end{pmatrix}$$ with $\nu$ a fixed nonsquare.  There are $q^2$ unipotent matrices in all (special case of a general result of Steinberg), half in each class for $p$ odd.   The usual canonical form theory just has to be adapted a bit over a finite field.
4) If you also want to look at elements of order 2 when $p$ is odd, you have to work a little more.  Again it's probably easiest to determine (as in Dornhoff) all classes of semisimple, unipotent, or mixed elements, along with the sizes of their classes.  It's not clear to me what motivates the format of your question, however.
ADDED: To be more explicit about elements of order 2 when $p$ is odd, just look at the standard list of classes in $G$ and note there is a unique one containing elements of order 4 (hence order 2 in the projective group).   Such elements are semisimple, but may be diagonalizable over $\mathbb{F}_q$ or not depending on which of $q-1$ or $q+1$ is divisible by 4.