Proportional Representation (PR) describes various election systems which try to ensure that the proportional support gained by different groups is accurately reflected in the election result.
In practice this usually involves ensuring that political parties in parliament or legislative assemblies receive a number of seats (approximately) proportional to the percentage of vote they received. This is known as party-list proportional representation.
Another kind of election system that strives to achieve proportional representation but which does not rely on the existence of political parties is the single transferable vote (STV).
Some systems, such as the single non-transferable vote and cumulative voting are described as semi-proportional.
Systems that do not result in proportional representation are known as majoritarian systems. These include the single member constituency/plurality, also called the first-past-the-post, the single member constituency/ majority, also called the majority run-off, the alternative vote, and the bloc vote, in which parties can receive seat numbers that bear no relationship to the national percentages they received in parliament.
The district or constituency magnitude of a system (i.e., the number of seats in a constituency) plays a vital role in determining how proportional it can be. When using proportional systems, the greater the number of seats in a district or constituency, the more proportional an electoral system can become. Any system with single-member districts is by necessity majoritarian at district or constituency level. However, district or constituency borders may be gerrymandered to create "majority-minority" districts or constituencies where a group of voters in the minority system-wide form the majority in a particular district or constituency, thus allowing a simulation of proportionality system-wide.
However, multiple-member districts do not ensure that a system will be proportional. The bloc vote can result in "super-majoritarian" results in which, in addition to the normal disproportionality of single-member majoritarian systems, geographical variations that could create majority-minority districts become subsumed into the larger districts.
Proportional representation seems unusual to Americans, but it is actually a much more common system of voting than first-past-the-post. In general, first-past-the-post is only used in former British colonies, and even Britain itself uses PR for the Scottish and Welsh assemblies and for its EU delegation. Nonetheless, PR does have some history in the United States. Many cities, including New York, used it for their city councils as a way to break up Democratic party monopolies.
Some electoral systems incorporate additional features to ensure absolutely accurate or more comprehensive representation, based on gender, ethnicity, etc. Note that features such as this are not strictly part of proportional representation; depending on what kind of PR is used, people tend to be already represented proportionally according to these standards in a natural way.