In Portugal we have a President and a government lead by a Prime Minister that usually is the leader of the party with most votes, be it a single party government or a coalition govenment. The leader of the most voted party does not have to be the Prime Minister, for government, we vote for a party, not a person, it can be another person from the same party.
The voting system for government is some how similar to Canada's one. The larger administrative parts Portugal is divided are Distritos, each with representation on Parliament, including the autonomous regions of Madeira and Açores (Azores) - Madeira and Açores also have their own local parliements.
Like in Canada, we have several parties and we have had coalition governments. Our current government is a coalition one, and, actually, the main party in the coalition was not the most voted party. The most voted party did not had enough votes to govern nor could find anyone to make a coalition with them. In a case like that the next most voted party, if able to form a coalition can govern, if not, new elections will be held.
Presiodential elections are separated from government/parliament elections. The President can be from one party, or an Independent, and the Prime-Minister from another. Currently we have a President that is a member of a party, but has run as an Independent, a Prime Minister from another party and a government with three parties. There are more three or four parties in Parliement.
If the Prime Minister dies, another member of his party, be it a cabinet member or not, will become the Prime Minister, if something happens to the President the sucessor is the head of Parliement. We do not have a Vice-President.
We call Minister to what in the US roughly matches our Foreign Office Minister. State Secretary is a position below minister, either from the same Ministry or for things that do not have a Ministry (those things depend of each government, sometimes Culture is a Ministry other times it is not), there are also Under Sub Secretaries, a position below State Secretary.
The number of ministers and ministries depends of each government.
Seats in Parliement are proporcional to the votes each party had, but Parliement is a different body. The bigger party may not agree with all policies from its government.
We also have "motion of non-confidence". The President can dissolve Parliement under certain situations and new elections will be held.
Spain, that like the UK is a Monarchy, has had three elections on the space of 18 months, I think, because the most voted party did not had a majority and the other parties refused to made a coalition. Third time around they allow that party to rule as a minority. A minority party will have to strike deals in Parliement and is always under a lot of stress.
France system is more like the US one, the President runs the country, but they have a Prime Minister. Of course they also have a parliement. Germany has a chancellor, and a President. People only ever heard of the Chancellor, usually, no one outside Germany knows the name of the German President. Germany also has a parliement, called the Bundestag.
We call our Assembleia da República (The Republic's Assembly).
Several countries in Europe are monarchies, the King/Queen is the head of state, but the country is de facto run by the government. Kings and Queens are not electec, so like in the US, they only have a major national election.
Sometimes good things fall apart so better things can fall together. (Marilyn Monroe)