The labels ‘left’ and ‘right’ in politics originated in the first French National Assembly in 1789 following the revolution, and was named according to the seating arrangement of the groups in the chamber as seen from the president’s chair. The largely economic views associated with those groups became synonymous with left-wing and right-wing politics, with the left seeking social justice largely through redistributive social and economic policies.
One of the things I find most annoying about political discussions in the US is the idea that the struggle here is between the left (as represented by the Democratic party) and the right (the Republicans). To describe it as such is to drain those political labels of any historical and political meaning and make them merely synonymous with two ends of a very narrow spectrum.
When measured against its historical meanings, what the US has is a center-right party (the Democrats) and a radical right wing party (the Republicans). The idea that Barack Obama is a radical left-wing socialist is to distort the political and economic tradition of socialism by tarring it with the kinds of pro-oligarchic policies that he advocates.
Of course, any simplistic left-right split based purely on economic points of views is somewhat simplistic and does not capture the full richness of a person’s views. This website adds another axis that measures where one stands on the authoritarian-libertarian axis in addition to the left-right social/economic axis and provides a questionnaire to measure where people stand. You can do the test in a few minutes.
I came out fairly deep in the bottom-left quadrant (6.5 towards the left, 6.5 down on the libertarian axis), suggesting that I was pretty left wing and libertarian, which did not really surprise me.
Update: You can compare yourself with some world leaders from a few years back.