Suppose that in a republic with representative democracy, ie. a form of government that has a so called "parliament" where parliamentary representatives, who are voted into that position by people, have the people's mandate and power to decide on legislation, there's a parliament of, say, 100 representatives.
How are these 100 representatives selected for parliament, after an election?
One would think it's a pretty straightforward thing: Just take the 100 candidates who got the most votes, and put them into parliament. That only makes sense: Select the people who got the most votes. How else could it be, duh?
The problem is that that kind of system doesn't work, and would rather ironically go against the principles of democracy and majority representation. How so?
Well, suppose there's a really popular candidate who represents the will of 90% of the population, and thus receives 90% of all votes. The remaining 10% of voters all vote for people who represent the opposite positions.
What happens if we just put the top 100 candidates into parliament? What happens is that there would be 1 person representing the majority opinion, and 99 people representing the opposite opinion. Which of course means that the 1 representative will never get that position into law, as he will be outvoted 99-to-1 by the opposition. Which means that 10% of the population will rule over the remaining 90%.
That's not how democracy should work.
That is the reason (or one of the main reasons) why parliamentary elections in most countries tend to be a lot more complicated than just "put the top n candidates into parliament". The more complicated systems exist to try to make sure that the will of the majority is actually represented in parliament, in the proportion that was voted. In this hypothetical example, a fair election system would try to make sure that 90% of parliament represents that majority opinion and 10% the opposite. In other words, it tries to make sure that the split is 90 parliamentary representatives (give or take) represent the majority opinion.
Which, of course, means that people who didn't get the top-100 votes will end up in parliament. Sometimes even people who didn't get any votes at all! And, conversely, it means that some candidates who were in the top-100 will not get into parliament. This on its own sounds a bit crazy, but such systems exist to make sure that the will of the people is actually represented in a fair proportion.
In small-enough countries the head of state (most commonly "President") is often elected by directly counting all votes, and the winner gets elected.
In larger countries, however, this can also pose a similar problem: In large countries there tend to be quite diverse areas, which have different interests and different population amounts. A direct vote count would give undue power to those areas with most population over those with less population. Those areas with more population might have rather different interests than the others, and those interests might in fact be detrimental to the smaller areas.
Thus, some big countries, like the United States, implement a more complicated system of voting for the head of state, which tries to ensure that the actual will of the entire country is represented, rather than giving larger states power over smaller ones.
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