The modern theory of vector bundles over a curve starts with Grothendieck's article *Sur la classification des fibrés holomorphes sur la sphère de
Riemann. American Journal of Mathematics, 79, 121–138, 1957.*  
(Actually Geyer and others  afterwards  realized that the gist of the theorem, in another formulation of course,  goes back to results of Dedekind-Weber back in 1882! See Scharlau's article below)    
Atiyah then classified, also in 1957, vector bundles on an elliptic curve in [this](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CC0QFjAA&url=http%3A%2F%2Fwww.math.lsa.umich.edu%2Fseminars%2Falggeo%2FtopicsAG%2F414.pdf&ei=b0awUa_pMe6o0AX8uYBY&usg=AFQjCNFkzBF3Fh7v6Zl4M1UDWTUN80tr-A&bvm=bv.47534661,d.d2k)  splendid article.   
There was then in the next decades an intense activity in gettting results on moduli spaces for curves of genus $\geq 2$.   
Leaders in the field were among others Narasimhan and Seshadri and here too one can find  older predecessors, notably André Weil with his  1938 article article *Généralisation des fonctions abéliennes*.  

**Bibliography**   
Some pleasant didactical references :  

For the classification of vector bundles on complete curves in the spirit of Geometric Invariant Theory there is Chapter 5 of [Newstead's book](http://books.google.fr/books/about/Lectures_on_introduction_to_moduli_probl.html?id=1yAiRAAACAAJ&redir_esc=y)     

[Here](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=5&cad=rja&ved=0CEgQFjAE&url=http%3A%2F%2Fwww.mcm.ac.cn%2Factivities%2Fpapers%255C2009-9-18-Sui-3.pdf&ei=rUuwUdiVD6Kk0QXfnIHYBw&usg=AFQjCNE5YXFbvVE0h4kmliGuShT8I1fYaQ&bvm=bv.47534661,d.d2k) is a great survey by one of the creators of the theory.    
 
[Scharlau's history](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CDIQFjAA&url=http%3A%2F%2Fwwwmath.uni-muenster.de%2Fu%2Fscharlau%2Fscharlau%2Fgrothendieck%2FGrothendieck.pdf&ei=t0ywUcWNE8i-0QWMv4DYCg&usg=AFQjCNF19wUugOA-52Q9s-VH_w00bAflUg&bvm=bv.47534661,d.d2k) of Grothendieck's classification. And a proof  of that classification  can be found on page 23  of  [Montserrat Teixidor's](http://www.tufts.edu/~mteixido/files/vectbund.pdf) survey (which by the way is one of the best texts I can recommend as an answer to your question).  

An [introduction](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&ved=0CC0QFjAA&url=http%3A%2F%2Fwww.uiweb.uidaho.edu%2Fjohnsonleung%2Fresearch%2Fvbec.pdf&ei=w1SwUdPuDO-A0AW_kIHYAw&usg=AFQjCNEt_f32HcjGfyP_E8ovXadPzTSWxQ&bvm=bv.47534661,d.d2k)  to Atiyah's classification.  

And finally, [another survey](http://www.google.fr/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&cad=rja&ved=0CDcQFjAB&url=http%3A%2F%2Fwww.cimat.mx%2FEventos%2Fc_vectorbundles%2Framanan_notes.pdf&ei=glSwUYWMCa2l0wXz-4D4BQ&usg=AFQjCNE95b7l9q7SGV1uqdbK6uKKh0H-yA&bvm=bv.47534661,d.d2k) by one of the historical masters of the field.