In a proportional representation system, parties generally get to be represented in the legislature in proportion to the amount of the vote they receive. In other words, a party that got 25% of the vote in a US election would get 0% of the representation. In countries like Germany that use PR, that party would get around 25% of the seats in the Parliament.
Some argue that this system is fairer than the US system. It allows more parties to enter politics and thereby lets more people have their voices heard. In the US, a large minority might not be heard because it cannot defeat the major parties. In a PR system, it would be heard.
The flip side to this is that a PR system is more prone to extremism and parties that concentrate only on single issues. A party can get into the government and "cause trouble" simply by having strong support on a single issue. Parties in PR, unlike those in the US, do not have to appeal to a broad range of people and so they do not encourage compromise.