What are the merits of different electoral systems?

I was a difficult question this time, it took me few hours to sort it out. So there you go:

The first-past-the-post system: the second ballot and the alternative vote. It is two round voting system used to elect a single winner. In the second ballot system the voter simply casts a single vote for her or his favorite candidate. In presidential elections, if no candidate receives an absolute majority of votes, then all candidates, except the two with the most votes, are eliminated, and a second round of voting occurs. For example it took place in France in 1995. In the alternative vote voters rank a list or group of candidates in order of preference. In other words (on the most common kind of ballot paper used) the voter writes a '1' beside their first choice, a '2' beside their second preference, and so on. In single-winner elections a preferential ballot paper can be redistributed according to the voter's expressed preference. Votes allocated to a candidate that is to be excluded from the election counting process are redistributed and allocated to the subsequent continuing candidate in order of the voter's indicated preference.

Strengths: easy to understand; executive strength; an aid to party unity; enhancement of the links between the citizens and legislators.

Proportional representation: the single transferable vote; the party list system and the additional member system.

The single transferable vote initially allocates an individual's vote to their most preferred candidate, and then subsequently transfers unneeded or unused votes after candidates are either elected or eliminated, according to the voter's stated preferences. To be elected a candidate has to secure a quota of votes. This system applies to legislative body elections.

The party list system is used in multiple-winner (elections to parliament for example). It ensures proportional representation. This system can be also part of the additional member system (some representatives are elected from geographic constituencies and others are elected under proportional representation from party lists). In these systems, parties make lists of candidates to be elected, and seats get allocated to each party in proportion to the number of votes the party receives. Voters may vote directly for the party, as in Israel, or they may vote for candidates and that vote will pool to the party, as in Turkey and Finland.

Advantages: minorities are fairly treated; parties are stimulated to cooperate what helps to avoid political extremes.

Liberal democracies use a variety of methods to elect candidates for public office. Both systems (the first-past-the-post system and proportional representation) are different, but both are used. For example, in Russia both methods are used in legislative process. The first system has many strong points while its weak points are eliminated in the second system. Still, both are not perfect.