Westminster is what many Commonwealth countries derive their government structure from. Canada probably has the most traditional version of Westminster next to, well, Britain inasmuch as we don't have an elected Senate (appointments ala the House of Lords). Australia, for example, is also a Westminster democracy but has elected to adopt American-style systems including a bicameral system where both houses are elected (and in fact, BOTH houses have power over the budget) and, curiously, an independent state that houses their national capital.
In practical terms, this means we have a head of state, the GG whom represents the Queen/King, and the parliament answers to the GG. It's why, for example, the government has the GG read the budget to the public among other things.
Historically, in Britain, it was a system of governance created to allow for the crown to remain in defacto power, even if all the actual power is given to the people - in this case, represented by parliament.
I actually don't know how the Nordic countries, who also have sitting royalties, (or Japan for that matter) function, but I imagine that it is mostly the same. Arab countries seem to offer more power to their royalty (Jordan, for one), but I really don't know anything about how their governments are run either.
The other popular system of government is the Republican system. This is what the French and the Americans use, in that there is a directly elected executive who is the actual head of state and replaces the royalty. France, obviously, had their issues with their own royalty (to say the least) and America's existence is born out of not wanting to be subjects of the crown.
In terms of various forms of electoral systems, very few democratic countries actually use First Past The Post, which is exactly what it sounds like - the person in first place, regardless of how many votes are received, is declared the victor.
Most other democracies are use proportional representation of some sort (you vote for a party or a slate of candidates, not for an individual candidate) or preferential ballot voting (you rank your candidates from first to last, and only when someone receives 50% of the votes are they declared the victor - this is how party leaders are elected in Canada and the United States, for example).
I suppose every system has their advantages and disadvantages, but it's mostly academic since the last two attempts to change the electoral process in Canada ended in failure and we will definitely never break away from the Commonwealth any time soon.
Oh right, systems of governance are mutually exclusive with systems of voting. As mentioned above, Australia is also a Westminster style democracy, but they do not use FPTP to elect their members of parliament or senators.
