There are two main types of voting systems that can lead to different party systems. These are “first past the post” or “winner take all” systems on the one hand and proportional representation systems on the other. Each of these will tend to lead to a different type of party system.
In a winter take all system, only the party that gets the most votes gets any representation. For example, a party can win 45 percent of the vote can lose the election and end up with no representation. This does not seem fair because that means that 45% of the people do not get to have anyone representing them. This sort of system will typically lead to a two party system like that of the United States. That is because only parties that can actually get really large numbers of votes will ever get any representation.
In a proportional representation system, every party that gets over a certain percentage of the vote will get representation proportional to their voter percentages. That is, a party that gets 30% of the vote will get more or less 30% of the representation. This leads to a party system with many small parties because parties can be rewarded even if they only get small portions of the vote.