In its present form the question is not clearly enough formulated to have a definite answer (as Marc Palm points out in his comments), so it's difficult to upvote. What I can do is point to some of the relatively old material which in principle should provide answers, though most of it is not readily available online.

1) It's natural to begin with centralizers of semisimple elements in a connected semisimple algebraic group $G$, then adapt to finite groups of Lie type. Here you can start with a connected reductive group, with no loss of generality, but in fact the centralizers depend just on the study of a semisimple group following Borel-Chevalley, Borel-Tits, Steinberg, Carter. Here the "simple" groups (in the sense of algebraic groups) are most crucial, and at some point you get involved with case-by-case study. For the finite groups, it's most efficient to work over an algebraic closure of a finite prime field, then study split and quasi-split groups along with groups of Suzuki and Ree.

2) Semisimple elements $s \in G$ are the best behaved ones, having reductive centralizers. But the centralizer need not be connected, unless $G$ is assumed to be *simply connected* (a difficult theorem proved by Steinberg with help from Springer and still awaiting a more transparent proof). Otherwise life gets more complicated, as seen in the format of character tables for the finite groups in the *Atlas*. (The Weyl group may contribute a little extra to the centralizer.)

3) Now the structure of $C_G(s)$ is transparent: it is connected and reductive, generated by a maximal torus containing $s$ along with root subgroups relative to this torus which belong to roots lying in a subsystem generated by roots corresponding to a proper subset of vertices of the *extended* Dynkin diagram (Borel de-Siebenthal theory, applied by Carter and his student Deriziotis). In particular, $C_G(s)$ may or may not be a Levi subgroup of some parabolic subgroup of $G$, though it is always a *pseudo-Levi subgroup*. For instance, in type $G_2$ you can have such a subgroup with derived group of Lie type $B_2$.

4) By the time all of this gets adapted carefully to rational points of groups which are split or quasi-split over a finite field (or groups of Suzuki, Ree types), the details multiply. Carter and Deriziotis organized much of this material in their papers and lectures, referenced in my 1995 AMS book *Conjugacy Classes in Semisimple Algebraic Groups* (with corrections posted on my homepage): see especially Chapters 2 and 8.