US history is interesting. The Republicans and Democrats of today do not hold the same beliefs as those who belonged to their party prior to 1930 or even 1860's.
Republicans are typically described as "Conservative" either fiscally, socially, or both whereas Democrats are "Liberal" in both. Unfortunately people associate the word "liberal" with Democrats which is a shame because it doesn't always mean such (Ex. www.Mises.org
The press associates the Republicans as being favorble to Big Business and Wall Street however some of the biggest supporters of both are Democrats (Senator Charles Schumer, Senator Chris Dodd, Congressmen Barney Frank). The press also associates Republicans with most Protestants groups under the idea of family values (Makes them easy targets when they cheat on their wives).
The press associates the Democrats with Labor Unions, Environmentalists, Technology (Silicon Valley is in California which is a "liberal state".), Non-whites or people of atypical sexual persuasion.
The reality is that these are generalizations and both parties don't always function according to their CORE principles. (Ex, Republicans have always campaigned on fiscal restraint when it comes to spending money yet the Iraq and Afghan campaigns have been financed 100% with debt. From ~2000 to 2008 the Republicans had a majority within both the House of Representatives and the Senate as well as a Republican President.)
The United States is a republic not a democracy and it has been trending socialist for a long long time. It's more of a corporate oligarchy where the people still vote but the candidates all come from the same pool.
I've always felt Ayn Rand has the best description of both groups (you've already read that post).
IMO the last Republican is Congressmen Ron Paul.
There really are no european equilivants. I think in Europe you have groups which carry some ideals of both the Republicans and Democrats. Something to keep in mind is that the United States IS the largest consumer of goods and services within the world and a superpower. This means that most European countries rely on our consumption to fuel their economic growth. This and the defacto protection allows them to spend more $$ on social programs than say defense.