A multi-party system in politics is a system of government in which more than two political parties truly have a chance to get real political power. This means that more than two parties have a chance to either govern on their own or to be part of a coalition government (as the Liberal Democrats and the Tories are part of a coalition government in England right now).
Multi-party systems are different from one-party and two-party systems. A one-party system is like China's -- it is basically not a competitive democracy because only one party has any chance at power. A two-party system is like that of the United States. There can be many parties (as there are in the US) but only two of them are relevant and can hope to gain power. By contrast, in a multi-party system (such as that of Germany or Israel) many parties can hope to have some share of the power.
In a multi-party system, there are multiple political parties capable of taking charge of the government. Since the parties tend to be well structured and separate, they also tend to represent either different ends of the political spectrum or specific issues relevant to the country in question. A multi-party system also speaks to the method of transition between the controlling parties. In the United States, the two main parties are the Democratic and the Republican parties. The transition of power then is through the election process that, for the presidency, takes place once every four years.