Proportional representation is a system used to elect a country's government. This means the results of an election decide directly how many seats each party has got. Decisions are then made by the people who are elected. This has advantages over other systems, and some disadvantages.
The alternative is the first-past-the-post system in which constituencies vote in one member, and that's all there is to it. That is how it is in the United Kingdom.
In most western countries, there are more than one political party. Each elected representative will be a member of one or another party. If one party has an overall majority, then it forms the government. Otherwise the government must have members of more than one party.
Countries which use proportional representation include: Austria, Argentina, Belgium, Brazil, Bulgaria, Czech Republic, Denmark, Estonia, Finland, Greece, Iceland, Ireland, Israel, Netherlands, Norway, Poland, Portugal, Romania, Russia, South Africa, South Korea, Spain, Sri Lanka, Sweden, Switzerland and Turkey. But not the United Kingdom.
Countries which have systems that are similar or use semi-proportional representation include: Australia, Germany, Hungary, India, Republic of Ireland, Italy, Japan, Mexico, New Zealand, and Thailand. India is one of the most successful examples of a country with proportional representation.
Similar principles apply to sub-regions, who may have their own parliament or assembly.