From Academic Kids
A multi-party system is a type of party system.
Unlike a single-party system, it encourages the general constituency to form multiple distinct, officially recognized groups, generally called political parties. Each party competes for votes from the enfranchised constituents (those allowed to vote). A multi-party system is essential for representative democracies, because it prevents the leadership of a single party from setting policy without challenge.
If the government includes an elected congress or parliament, the parties may share power according to Proportional Representation or the First-past-the-post system. In Proportional Representation, each party wins a number of seats proportional to the number of votes it receives. In first-past-the-post, the electorate is divided into a number of districts, each of which selects one person to fill one seat by majority (or plurality) vote. First-past-the-post is not conducive to a proliferation of parties, and naturally gravitates toward a two-party system, in which only two parties have a real chance of electing their candidates to office. (This effect is known as Duverger's law.) Proportional Representation, on the other hand, does not have this tendency, and allows multiple major parties to arise.
This difference is not without implications. A two-party system requires voters to align themselves in large blocs, sometimes so large that they cannot agree on any overarching principles. According to some theories, this allows centrists to gain control. On the other hand, if there are three major parties, each with substantially less than a majority of the vote, two of them may find it necessary to compete for the support of the third. Some argue that this gives the third party inordinate political leverage.