How do different types of voting systems lead to different types of party systems?

In a Presidential system that uses first-past-the-post, it is hard to get more than two or three significant parties. Simply put, a party that only pull 15-20% of the vote will never win the election and, over time, its supporters will transfer their allegiance to one of the parties that is competitive.

In a parliamentary system that uses first-past-the-post, the same rule applies for national parties, but regional parties can make an impact, possibly holding the balance of power when neither national party can get a clear majority. A good example of this is the United Kingdom and Canada where you have three national parties and also parties that are competitive in different regions (Bloc Quebecois in Quebec, Scottish Nationalists in Scotland, and Plaid Cymru in Wales.)

Multi-member systems (either proportional representation or single transferrable vote) results in a larger number of viable political parties that win seats in the governing body. A good example of this is Germany where the balance of power is held by the second tier parties that win seats through the party list not through individual constituencies.