Yes, there are many such countries in Europe and they function very well.
Multi-party democracies are generally run by coalition governments. Usually, not a single party holds more than 50% of the seats in parliament as needed for decision-making. Therefore, two or more parties cooperate in a government, as many as are needed to reach the treshold of 50% of the seats. What parties can form a government depends on their programmatic differences. They might negotiate cooperation before, or only after the elections.
For example, in The Netherlands, people vote for a parliament consisting of 150 seats. At the Elections of 12 September 2012, 11 parties got at least one seat in parliament. The smallest party had 2 seats, whereas the largest two parties had 41 and 38 seats, respectively. By comparison, in the 2010 elections, the largest parties had 31, 30, and 24 seats, respectively. Not any combination works, because parties may have great difficulties to agree programmatically. In the Dutch system, the largest party gets the head of government and gets the initiative to form a government. After the 2012 elections, it took 54 days for the two largest parties (centre-left and centre-right) to form a coalition government. After the 2010 elections, it took 127 days to form a three-party right-wing government. During the negotiations, the previous government continues as a caretaker government.
A coalition government may be less stable than a single-party government, because as new events come along, parties need to agree on how to take care of it. For example, the coalition government after the 2010 elections broke down when they could not agree on how to fix the budget when there appeared a need for massive cuts, which was why new elections were needed already in 2012.
Negotiations for coalition governments sometimes take a long time, because parties may be reluctant to give up electoral promises, particularly so soon after the elections. Yet this is often necessary, meaning there will always be disappointed people. This can be seen clearly in what happened to the two coalition parties in the opinion polls immediately after the government was formed:
Dutch polls from peil.nl. The title reads "Development of political preference in 2012"
The elections were in September, just after the big rise of the social-democratic PvdA (red dashed line) and the big drop of the socialist SP (red solid line). The coalition programme was announced late October, just before the big drop in the liberal (in the European meaning) VVD. It can be seen that the publication of the coalition agreement led to a big drop in VVD and PvdA (the coalition partners) and a major rise in the nationalist-populist PVV (grey solid line) and a rise in several other parties.
This example is from The Netherlands, but most European countries are multi-party parliamentary democracies with similar (but inequal) systems. For example:
Sweden has a 8-party parliament, but 7 of those (all but the extreme right) are divided in two major blocks, so it's effectively close to a two-party system.
Switzerland has many parties, but its governing coalition hasn't changed for decades (perhaps because everything important gets decided in referendums).
Belgium has the additional complication that parties need not only agree between left and right, but also between Flanders and Wallonia, leading to a record-breaking 541 days formation in 2010-2011.
There are many more examples.
An entirely different question is how the president gets elected in such a system. This is independent of the question of the government, but usually such systems don't have a president as strong as the one in the US. For example, in Germany the president has mainly a ceremonial function and is elected by parliament, and Switzerland doesn't really have a head of state at all. France does, and in France the president and government may not be of the same political colour, and consequently be constantly fighting.
So, certainly multi-party democracies exist and are functioning to varying degrees. Stability may be less than in two-party systems, but that's not always a bad thing; it's relatively uncommon to see the same people in power for more than 8 years, new parties have a chance to introduce themselves into the political system, and minorities are represented when elections are proportional. In summary, it has many advantages.