I took my Politics class from Aug.-Dec. this year (as well as EVERY semester in my H.S. years) but my college one is the only one that discussed European politics, so I'll try to impart as much knowledge as I can.
Europe (more focused on England/Great Britain):
They are broken up into various cities/regions instead of States/counties/cities/towns like the U.S.. This makes their govt. more centralized to their capital instead of 1 Federal with many State governments. Their votes elect the House of Commons M.P.s (members of parliament, should be parliament members to reduce confusion with military police, just saying...) the lower house, but with more people in it. The second house is the House of Lords/Nobles; they are of nobility and are of bloodline (like the King/Queen; one member's great, great grandfather might have saved a nobleman in the 1600's and thus got into the house of Lords, making his descendants automatically put into the position. I am unsure if it still is only males, or how it replaces dead bloodlines). They are of far fewer number and also help in the decision making process. There are some quirks in this however, as they (I believe this is required, but I am unsure of this as it may be a stereotype) wear robes and wigs to there. There are also more physical fights in Parliament than in the U.S. govt. (I don't recall any fights at all here) resulting in all out brawls at times (literally).
The head of their govt. is the Prime Minister, or the leader of the current majority party. This can get complicated in a 3 or more party system (as it is more common in Europe) and if a new majority is in power, the current P.M. is ousted (even if he is doing a great job) and the new majority's P.M. is in. This means that laws can change MUCH more quickly than in the U.S. system since the majority has much more sway in all level of government (Bush would be out in '06 if we had their govt., as an example). There is more, but I cannot think of it at this time, if other Europeans could add input I will be greatly obliged. I'm sorry that I can't be of more help in this subject.
Summary: More centralized in 1 location and has a much quicker and abrupt change in power, law, and party which is both a boon and a hindrance. England is unique in the uniforms and brawls.
Also, tell me of the different parties and their leanings (liberal/conservative)
U.S.:
Has 3 levels of govt. that all have a check/balance system on each other so that no one is more powerful than the other; however it has been of late that the Executive branch of govt. has been receiving more power since the President has become more of a law-man over a figurehead/military leader. The levels are as follows:
1. Legislative: Is composed of 2 parts, like Europe's, called the House of Representatives (Congressmen or Representatives) and the Senate (Senators). They make laws and appoint the Pres. cabinet and overturn vetoes. Also appoint judges. Lobbyists (private groups that fund Congressional campaigns to help their interests, have been downplayed in recent years) are a part of this as well (many are former members of Congress). They get many perks (free mail, no slander while on the floor, free travel, long with other things that I cannot recall at the moment).
A. House: They represent the majority of the population (balance toward the larger States) and are based on the total fraction of the population in a current State (3,500 per or something along those lines) and is capped at 435 members (1938, or today would be well over 2,000 congressional people). Each Congressperson represents a district in their State and deliberates on issues on the House floor. The Speaker of the House is whomever is with the majority party. Also, is 3rd in line.
B. Senate: They are on a 2 per State basis (power toward the smaller States), and represent the State as a whole (usually there is a rural/urban split like how Obama was the urban Northern half and (forgot name) was for the Southern farming half to deal with issues in a more localized fashion). There are 100 Senators total (2*50=100) and the Senate floor is more personal with no limit on speaking time (filibusters are where they can talk on end to kill a bill, but can be shut down by a maj. vote) and is more formal in the process. The House and the Senate combined make up the State's total Electoral votes for the Pres. election (a flawed system where only 12 States=victory) that takes into consideration the population and simplifies it into 540-something point system in a winner take all strategy (like Republicans, Democrats use a % system. 2 States do this as well, don't know who though). The leader of the Senate is the V.P. and is solely a tie breaker. There are other positions (whips, Maj./Min. leaders, etc.) that is too elaborate for this.
To see how a bill is decided (very funny and nostalgic):
http://www.youtube.com/watch?v=mEJL2Uuv-oQ
2: Judicial:
These are the 9 judges that serve for life (until retirement) and must be appointed by the President and approved by the Senate. They can declare laws unconstitutional (Jim Crow laws, our dark ages). Watch the last 2 seasons of Boston Legal to see what goes into the process, it's AMAZING (also, B.L. is my all time fav. T.V. show, what's the Euro. equivalent?). They are also the final point of the justice system (district, appellate, Judicial in both State and Federal tracks). They review an incredibly small percentage of cases presented to them.
3: Executive:
The Presidential system. It houses the President, his advisors, the press core, his Cabinet (he meets with his advisors, these guys just run the departments) the Secret Service, the bureaucracy, etc. all reside here. The Pres. can veto a bill, sign it, or pocket veto (don't sign, but Congress runs out of time in session). He is the Commander in Chief (leader of the military)and along with the Joint Chiefs of Staff (the Generals) command all positions of the U.S. military in the world (93% of all countries have a U.S. base in them). He greets all foreign leaders and (I believe) diplomats as well having Air Force/Marine 1 (and no traffic jams). The Vice President (first was the Prince of Darkness, now it's a gaffe machine

)is a minor player and can assume power if the main Pres. is out of commission. If he becomes Pres. more than 1/2 way through, it IS NOT A TERM and could mean 10 years in office, but even 1 day before is counted as a term. The Pres. can be impeached (taken out of office by Congress) and must make a State of the Union address every year to Congress.
U.S. summary: There is MUCH MORE that I didn't cover here (line of succession, lobbyists and the financial flow of money also known as the "Iron Triangle", Judicial review, the election process, the inauguration, etc.. It is slower, but more "sure" in it's methods of deciding what is/isn't a law and is currently one of the most complex and laborious of governments. It is also one of the largest with an amazing history behind it (I am hoping, one day, to be part of it).
Summary of the 2 types: They're AWESOME despite their flaws.