What are some political systems that have multiple political parties

The most important political system with multiple political parties is a democracy: for example in the US the Republicans and the Democrats struggle for power.

In constitutional monarchy, such as in the UK, there can also be multiple parties, that rule together with the monarch (or alone and  the monarch is just a representative function).

Some forms of oligarchy could also see multiple political parties, although this case is rare.

Democratic and Socialist governments have multi-party systems.  Usually this happens because interest groups want to better representation and each party tends to support or back certain agenda or ideology.  There are greater chances for winning positions in government. Unfortunately this causes deep divisions in the country that the majority gets lost.  In some countries, it often leads to conflict.  Some parties form coalitions to strengthen their chances during elections.  Countries that used multi-party systems are the Philippines, Israel, Italy, Mexico, Taiwan, Sweden, Portugal, South Africa, Romania, Sweden and the Netherlands to name a few.