On the original Riemann-Roch theorem I think Riemann first stated and proved a part of the Rieman-Roch theorem on a compact Riemann surface.
And later Roch supplemented it.
I wonder what the original statements of the R-R theorem by Riemann and Roch were and how they proved them respectively.
 A: A partial answer. According to the Wiki entry, it was originally proved by Riemann and generalized by his student Roch. The following contains the original statements. It does not address the question of how they were originally proved, which is given in the references.
Roch's generalization first appears in: Roch, Gustav (1865). "Ueber die Anzahl der willkurlichen Constanten in algebraischen Functionen". Journal für die reine und angewandte Mathematik 64: 372–376, which can be accessed via the same Wiki article. 
A translation of Roch's original statement of the theorem is given in A History of Geometrical Methods by J. Coolidge at p. 218: 
Riemann-Roch: If a rational function of x and y on the on the Riemann surface $f(x,y)= 0$ have N poles of the first order at N given point, and $i$ linearly independent functions $\frac{\phi(x,y)}{\partial f/ \partial y}$ vanish there, these functions depend on $N - p + 1 + i$ constants.
Riemann's original statement is given in Riemann, Bernhard (1857). "Theorie der Abel'schen Functionen". Journal für die reine und angewandte Mathematik 54: 115–155.
Riemann: Eine Function von $x+yi,$ die in einem Theile der (x,y)-Ebene gegeben ist, kann daruber hinaus nur auf Eine Weise stetig fortgesetzt werden.
I will leave it to someone with a better command of German to translate this, but the translation of Roch above helps. Someone with a better command of the subject might say whether this is in fact an adequate statement of the idea of his paper, which is in (typically) largely narrative form. It appears Riemann intended it to be so. 
This paper is available freely at DigiZeitschriften $^1$ but of course it is in German. I think the German is manageable with online resources and probably there are German natives on this site who could easily help with a problematic passage.
Here is an updated account of the idea, which neither cites nor quotes the original, which apparently has been merged into more general ideas. Also the original proof Riemann gave was apparently flawed (Weierstrass showed this) so some revision was inevitable.
There is a thesis by Carmen A. Bruni on applications of RR to algebraic geometry, with a brief account of its history, on the first page of google results under Riemann-Roch. It also does not contain Riemann's original language. 
There is a very readable history of the Riemann-Roch theorem by Jeremy Gray which is accessible in PDF under "History of the Riemann-Roch theorem."
Another paper including an historical introduction that gives a statement of the original Riemann-Roch theorem as Theorem 1.2.1 can also be accessed as a PDF: The Grothendieck -Riemann-Roch Theorem by A. Ellis 
1--Links not allowed to this site. You can link to it from the Wiki article on the Riemann-Roch theorem.
A: There are links in the Wikipedia articles on the Riemann-Roch theorem and on Roch to the original papers in the Journal für die reine und angewandte Mathematik.
