Can you please help me explain the difference between First Past the Post and Proportional Representation voting systems?

First Past the Post (FPTP) and Proportional Representation (PR) voting systems have different aims, produce different sorts of legislatures and have different consequences for a democracy's party system. Let's start with First Past the Post, which is the voting system we use here in the UK. Under the electorate is divided in to 650 population-based constituencies. Within each constituency, each political party puts forward a prospective Member of Parliament to run for election. Voters then choose for the individual they want to represent them. When results are counted, the individual with the most votes is given a seat in Parliament in each of the 650 constituencies. The leader of the Party with the most seats in the House of Commons is then made Prime Minister. This system is aimed to produce stable, moderate governments that can command majorities in legislature and can therefore govern effectively. Smaller parties away from the political centre, (i.e. the Greens or UKIP) are disadvantaged because although they may win a significant amount of votes across the country, they may return a smaller proportion of seats because they failed to beat the bigger political parties in certain seats. The PR voting system seeks to address this problem by making the proportion of seats in the legislature directly representative of the proportion of votes won. Take the 2015 General Election as an example, UKIP won 12.6% of the vote share but only won 1 seat in the House of Commons because their support was too thinly spread across the electorate. The SNP, by contrast, won 4.7% of the vote share but won 56 seats because they vote was highly concentrated in Scotland. This allowed them to beat the bigger parties in individual constituencies. Advocates of the PR system would say this is undemocratic and unacceptable. In countries that use this system, such as Germany, smaller parties do better and have more of a say in the legislative process. However, governments often must be a coalition of parties and are therefore more restricted in being able to fulfil their manifesto promises to voters. This trade off between democratic representation and effective governance is key to the difference between these voting systems.