In the US, we had six party systems since the revolution. Each time we translation, we get two mostly new political parties which sometimes reuse the names of the defunct parties. Wikipedia has a good description: http://en.wikipedia.org/wiki/Party_system#United_States
you (GP) could also check out the XKCD of american political history. it has quite a few of the party names that people used throughout (which are entertaining, in a geeky kind of way):
Thanks you all, this has been very instructive. I realize I had a massive misinterpretation about left and right in the US.
That's because, in France, left wing has always been associated to humanism (humans first, economics then) and universal rights (the whole world deserve to have rights), while right wing is most about economics and nationalism.
From what I understand, US left wing is historically about equality for the "real USians" (kind of nationalism of the poors) while right wing, until getting own by far right in XXth's end, was for a very liberal society, be it in term of economics or in term of social rights.
This would explain a lot in the recent NSA reactions, where we were very shocked that the debate seemed to focus on whether or not US citizen were spied, rather than if mass surveillance was a problem at all.