How does First Past the Post work, and what are some of its advantages/disadvantages?

First Past the Post is the voting system used across the UK (although used in combination with the Party List System in Scotland!). In this system, each voter places a single vote for a candidate standing in their constituency, who will almost always be a member of one of the major political parties (Conservative, Labour, Liberal Democrats, Green…). To win in a constituency, that candidate only needs to win more votes than any other candidate – importantly, they do not need a majority. As such, a candidate can win on 30% of the vote, as long as all the other candidates got less than this (26%, 12% etc). The party who has the most candidates in Parliament is then able to form a government. 
As each candidate only needs to win more votes than any other candidate in that constituency, it is more likely to end up with one party able to form a majority government. Some have commented that this is both more politically and economically stable than other electoral systems. It also allows this party to more easily carry out their manifesto commitments and promises without either forming a coalition with another party (see 2010) or having to overly negotiate with other parties in Parliament (see 2015). First Past the Post also provides a direct link between a voter and a representative (as you vote in your constituency) which can be seen to offer fair representation.
However, it can also be argued that the lack of proportionality favours the ‘big two’ parties at the expense of smaller parties. This is because a candidate who gets e.g. 30% of the vote wins that seat, and the other 70% of the vote is effectively ‘wasted’. There is no proportional allocation of seats based on the vote a party gets overall. You could argue that this is a negative (smaller parties find it very difficult to win seats) or a positive (it keeps ‘extremist’ parties such as UKIP/BNP out of Parliament). However, it is clear that a majority government (has the most seats in Parliament) will often be formed against the wishes of the majority of voters: in 2015, the Conservatives had a majority of seats in Parliament with only 37% of the popular vote. This also means that as well as ‘wasted’ votes, some constituencies are considered ‘safe’ and have not changed parties in 50+ years, which can lead to voters feeling disincentivised (and not voting) or voting tactically (such as voting for the ‘lesser of two evils’).