Democracy (as understood in the Western concept of the word) has existed in all kinds of ways and institutional constructions. All of these constructions have included the idea that one of the fundamental rights of the citizens was to organize themselves in political structures conveniently called "parties". Gathering of people interested in the same ideas, as well as vehicles for the personal ambitions of some, parties have always been a fixture of democracy (and even of many non-democratic regimes). The European institutions recognized this too and parties exist at the European level, even though they have some difficulty to be recognized by the EU polity.
This said, there are fundamental differences between the party set up in the different democratic countries. In some countries, they are simple associations of citizens, with little more difference with a sport club than their objectives. In others, they are an integral part of the institutions, with a status well apart from other groups of citizens. Another major distinction between party systems is simply their number in a a given country. Some countries have essentially a two party system (to simplify, 'left vs. right"), some have a three party one and many if not most have a multiple party system. It would be extremely disingenuous to pretend that these difference have no effect on the political institutions and the political life of a country. On the contrary, these differences actually contribute to define the institutions and are in turn defined by them. But, unlike the constitution of a country, the party set up is rarely or never fixed. This can have tremendous and unexpected effects.
Let's take a couple of example. The USA, for instance, have a two party system, very rigid and highly integrated in the country's institutions. Parties there are in no way ordinary associations, but there is simply little political life outside of the Republican and Democrat parties. The United Kingdom recently evolved from a quasi two party system to a theoretical three party one. Although we could argue that the UKIP is a 4th player, it weighs too little in the country's institutions to seriously threaten the Tories, Labour and LibDem hold on power. France, Germany or the Netherlands are all multiple party set up, at least to a point. While the weight of the different Dutch parties can be reasonably considered as truly variable, France and Germany both have a very dominant right and left parties (UMP and PS in France, CDU-CSU and SPD for Germany). But even in France and Germany, smaller parties can and do actually play a role, would it only be the "king maker" one between the dominant right and left formations.
It is interesting to see that the current massive global crisis which has struck the Western world has affected the political life of these countries very differently, depending on their party set up. Netherlands has reacted to the crisis in a fairly classic way. The balance between the 15 various parties has been strongly modified, but in the end the power didn't change hands much. The crisis first brought a centre government (Labour, Christian Democrats and Conservative), then a slight sway to the right (Christian Democrats, Conservative and National-Populists) as electors blamed the first post-crisis coalition for its inefficiency. In general, it was a typical answer of a multi-polar system: it was easy to adapt because the multiple choices meant mutilple solution and easy adaptation to the new reality. The down side of this greater adaptability of course is that it gave access to power to a fringe party. This can typically not happen in a two or three party system.
In the USA, the strength or rather sheer weight of the two parties is so huge that a crisis can only make the power change from one party to the other. The crisis (amongst other factors) contributed to Obama's victory and did put the Democrats in power. However, their relative inability to tackle problems quickly and the fact that the crisis lasted longer than they thought did bring Republicans a victory in the House after only two years. It is fairly clear for an external observer that the US bipolar system has the advantage of stability. But it is also fairly obvious that it suffers greatly, as a downside, of its rigidity. Outside of the two main parties, there is no room for political power. One can have influence without them, but not exercise power. It also means that the US institutional system has too much inertia and is unable to give a speedy and adequate answer to a major crisis. The result of this is what we can observe on our TV screens: besides the Republican and Democrat parties, citizens organize themselves in a series of alternatives. Originally, the Libertarian organisation had this role, but we now see also the Tea Party, the Occupy Wall Street organisation, the Coffee Party, and so on. Obviously, the American institutions being what they are, these organisations can only exist in reference and interdependence with the two traditional parties. They are opposed to them, yet they are forced to ally with them to exist in the political landscape and not only in the streets.
While both set up have their advantages and disadvantages, I have to say that I find the US system extremely dangerous. When, in a democratic society, citizens feel so estranged towards traditional politic organisations that they have to get down to the streets to get heard, something can go very wrong. Democracy's principle is to give legitimacy to the government by having the citizens feel represented by it. The two party system inadequation, rigidity and slow answer to crisis put this democratic basis into jeopardy.