I've added my own circles to the map showing countries I am going to talk about.
Canada:
We used First Past the Post, or FPTP. In such a system, each area elects one member, and that member is simply the one who received the most votes.
France:
France uses a two-round system. It is similar to FPTP except a standard FPTP election is only the first round.
In the second round, the top two candidates advance, so that a winner can be determined who has the support of the majority.
Ireland:
STV is what is used here. In such an election, voters can transfer their ballots to other candidates by ranking their favourite candidates with a 1, 2, 3, and so on. Ireland uses multi-member electoral areas; this is contrasted with Australia which, while also using STV, only elects one member per area.
Because of the multi-member nature of the seats, and the usage of the Single Transferable Vote, Ireland's system is seen by many as a 'light' Proportional system, without the need to delve into full Proportional Representation.
New Zealand:
Like Germany, New Zealand uses Proportional Representation. In particular, they use a mixed system, where some members are elected in single-member seats, and others are elected on a proportional list.
In these systems, each party wins however many list seats they need so that the final result is proportional.
Iceland:
Iceland uses a districted system of Proportional Representation. Each district elects a number of MPs based on a PR ballot. However, Iceland (like many countries that use sub-national districts for electing their MPs) uses levelling seats. This means the final result is fully proportional and equal to the nation-wide vote total, or at least as much as is possible. One seat in each district works as a levelling seat.
The exact math for calculating the levelling seats is too complex for this cursory examination.
South Africa:
South Africa has the purest system, a single nationwide list, based on Proportional Representation, without any threshold cut offs. Election is based on pure math.
Greece:
Greece uses a top-up system alongside nationwide list PR. There are many ways to do this, but in Greece's case, whichever party wins the most votes, gets an additional 50 seats. This makes the top-up system a sort of nationwide FPTP worth 50 seats. Due to the math, a party only actually needs to win around 40% of the vote in order to achieve a majority.
Japan, Mexico, and Russia:
I'd encourage everyone to go back and look at the map now that you know what the various colours mean. You'll notice there are a fair number of Purple countries, countries like Japan, Mexico, and Russia.
These countries use Parallel Proportional Representation. Like top-up systems, the explicit rationale behind these systems (compared to pure PR) is to preserve the ability to win a majority. Putin, for example, chose to move to PPR from PR as an attempt to strengthen his ability to win majorities.
Japan however transitioned from a FPTP-like Block vote so PPR for a different reason. Japan had become a "one party state" while remaining a democracy, as the governing LDP won every election in a 40 year period. Finally, in 1993, an opposition coalition defeated the LDP. the 8 party coalition helped to bring an a PPR system as it enables opposition representation.
I will show some examples of how both of these work.
There is a simple math formula that can help you guess the results of a FPTP election. It has a high error margin, but works better than any other formula so simple. To do this you simply square the popular vote for each party, and the resulting total of numbers will be the share of seats won.
For this example I will be assuming that the Cat Party has won 47%, the Dog Party has won 36%, the Rabbit party has won 14%, and the Bird Party has won 3%.
The square of these numbers are 2209, 1296, 196, and 9 respectively, meaning we can estimate that in a 200 seat assembly, that the Cat Party would take about 59.5% of the seats, the Dog Party about 34.9%, the Rabbit Party about 5.3%, and the Bird Party, 0.2%
The results of a FPTP and PR election would thus look like this:
119 - Cat
70 - Dog
11 - Rabbit
0 - Bird
94 - Cat
72 - Dog
28 - Rabbit
6 - Bird
This puts us in a situation where we need to choose one of two options.
Option A - A Majority that has a weak Rabbit Party, and a Bird Party without any seats
or
Option B - A Minority that has a representative Rabbit Party, and a Bird Party with seats
Most people, when asked about electoral fairness, are actually alright with a party winning a majority on a minority of the vote; not only does it bring stability, but it also allows for focused policy. A party that won 47% should - in their minds - be allowed to govern on their own.
Most people, when asked about electoral fairness, want the smaller parties to have a voice. While there is a concern about "extreme" parties, when parties are actually named (IE the Canadian Greens) people are nearly universally alright with additional representation.
As such, compare these two outcomes with that of a Parallel system. 100 seats, picked proportionally, with no regard do what happens in the other 100 seats; and 100 seats chosen via FPTP.
107 - Cat
71 - Dog
19 - Rabbit
3 - Bird
Not only do we get a stable majority, but we also have representation for the smaller parties.
One of the great things about these systems is you don't need the same number of PR seats and FPTP seats. Russia assigned 225 seats to PR and 225 to FPTP. Mexico however assigned 300 to FPTP and 200 to PR. Japan assigned 300 to FPTP and 180 to PR. As such, you are able to adjust how proportional your system is, and, thus, how unlikely a majority is. Lets use the above example to show this in action.
With 50 FPTP seats and 150 PR seats, the majority is razor thin:
101 Cat
71 Dog
24 Rabbit
4 Bird
With 50 PR seats and 150 FPTP seats, the majority is far larger:
113 Cat
70 Dog
15 Rabbit
1 Bird
As such you can adjust your system, to be used in your country, province, or other area, to produce just as many Majorities as you desire.
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