A basically, hawk is someone who wants war, a dove is someone who doesn't.
Left and Right are the ends of the political spectrum. Left is "liberal" and right is "conservative" - that applies to the politics of just about any nation.
Here is a simple political spectrum. Ideally, democracy encompasses the egalitarian center between socialism and federalism.(X) Totalitarian states lie along the line outward from socialism to communism and from federalism to fascism.
Liberal and conservative should be within the democratic center of the spectrum. However, in american politics this is more and more distorted by neo conservatism which occupies the line somewhere between federalism and fascism. Neoconservatives equate liberalism with socialism and communism. The center then, tends to drift rightward.
You can forget all of that, except for HAVOC451's post...maybe. At this point every U.S. politician is professionally hawkish. The only difference is in degree.
As far as "liberal" and "conservative"? Your guess is as good as mine; the labels have been so corrupted over the years, especially "conservative", that there is no simple answer.
"The Hawk and Dove" is a nice tavern on capital hill in DC.
Hawks favor war, doves favor peace. One can be generally hawkish, or dovish, or, one can be a hawk on Iraq, and a Dove on Bosnia for instance. It is not a great system for labelling. One irony is that almost all generals oppose almost all wars. Should we really label them doves?
Left-wing and right-wing are generlly the extremes of liberal and conservative, respectively. It is a bit clumsy, as there are multiple facets to what makes one liberal or conservative. There are also big issues, like trade, which defy labelling.
Actually the Hawks and Doves were special to the Vietnam war. Although there are a lot of opponents of the Iraq war, and a lot of enthusiasts for it, I haven't seen the terms hawk and dove used recently.