> Technically a Prime Minister doesn’t win an election at all: people vote for a party.
In most parliamentary systems, the Head of State is compelled to invite the party with the largest representation in parliament to form a government, and that party then proposes a government which is then subject to a confirmation election in the parliament, which it must win in order to be the government. So, generally, Prime Ministers win an election among members of parliament, who are themselves elected by the public (whether or individually or by party name depends on the particular parliamentary electoral system -- the UK system, as I understand, uses first-past-the-post with candidate names and parties on the ballot, much like most US elections -- while some systems use party-list-proportional where seats are allocated proportionally to the number of votes cast for parties, and there are various hybrid and other systems in use, as well.)
This is not entirely unlike US Presidents, who are also elected indirectly by presidential electors who are in turn elected by the public, however, it does differ in that the electors in the UK are also legislators rather than being elected solely as electors. Its kind of like if, in the US, members of Congress elected the President.
(Which, in fact, can happen in the US system, but only if the Congressional count of the vote by the electors doesn't give a majority for any candidate.)
I'm not too familiar with the UK system, but it seems like in the US the people have more direct control over the identity of the president, since both parties have competitive primaries every election cycle, except for the parties of first-term incumbents, whereas the last Conservative Party leadership election was in 2005(?). Also, primary voters don't need to be paid-up members of their party.
The US differs from most European countries in that the president is also the head of the government. And because he's elected directly (well, sort of), the US government has its own mandate independent from the parliament, which means the parliament can't really send the government home (though it can clearly stop it from functioning, as the US has demonstrated).
In most parliamentary systems, the parliament is the direct representative of the people, and they elect and appoint the government, and can disband it whenever the government loses support of the parliament. The UK is slightly odd here, because I believe the members of the government are also still members of parliament. I have no idea whether the LibDems have the power to withdraw their support and send Cameron home (or at least back to his seat, while the LibDems form a new government with Labour). In many other countries, they would definitely have that power.
Of course the US and the UK are both oddballs in that they have a pure district system, whereas most democracies have some level of proportional representation. (Often purely proportional, but some have a hybrid system where people still have a representative directly from their district.)
In most parliamentary systems, the Head of State is compelled to invite the party with the largest representation in parliament to form a government, and that party then proposes a government which is then subject to a confirmation election in the parliament, which it must win in order to be the government. So, generally, Prime Ministers win an election among members of parliament, who are themselves elected by the public (whether or individually or by party name depends on the particular parliamentary electoral system -- the UK system, as I understand, uses first-past-the-post with candidate names and parties on the ballot, much like most US elections -- while some systems use party-list-proportional where seats are allocated proportionally to the number of votes cast for parties, and there are various hybrid and other systems in use, as well.)
This is not entirely unlike US Presidents, who are also elected indirectly by presidential electors who are in turn elected by the public, however, it does differ in that the electors in the UK are also legislators rather than being elected solely as electors. Its kind of like if, in the US, members of Congress elected the President.
(Which, in fact, can happen in the US system, but only if the Congressional count of the vote by the electors doesn't give a majority for any candidate.)