Australia
Federal Parliament consists of two houses: the House of Representatives (Lower House) and the Senate (Upper House). Election to the HoR is by preferential voting. Voters list candidates in order of preference: 1, 2, 3, 4, etc. If no candidate achieves an absolute majority on first preferences, then second perferences (then third, if still no majority is achieved) are counted. This system allows minority parties considerable influence: while they may not receive many first preference votes, distribution of their supporters’ second preferences can be vital to an election.
The party holding a majority in the HoR forms Government. The leader of this party is Prime Minister. The PM is a Member of Parliament like any other; the nation as a whole does not directly vote someone into this office. The PM is Head of Government but NOT Head of State. The minority party in the HoR is known as the Opposition.
The Lower House is dominated by two major political parties: Labor (note the absense of a “u”) and Liberal. The Libs currently hold Government in coalition with the National Party. Labor may be vaguely characterised as “left” and Liberal as “right”, although both have numerous factions within the party with widely differing points of view. Note that positions held by the nominally “right”/conservative Liberal party would be viewed as “leftist” in American terms.
Election to the Senate is by proportional voting. Distribution of voting districts is uneven: each Australian state regardless of population has 10 Senators (the Territories have two). The Senate is dominated by the major parties above, but the proportional system of voting has allowed minor parties to grab the balance of power. Democracts, Greens and independents have considerable influence in the Senate: at present, their cooperation is vital to the passage of Government legislation opposed by Federal Labor.
The Senate was contrived as a house “of the states” (hence 10 Senators from each), but it is drawn-up along strict party lines, like the Lower House.
Australia’s Head of State is the Queen of Australia, Queen Elizabeth II. Her power is exercised in her Australian representative, the Governor-General. However, the Queen has no real power in Australia: the G-G’s powers are mostly ceremonial and the “reserve powers” are exercised only on “advice” of Government (“advice” meaning direction.)
Executive power is vested in Cabinet, made up of the Ministers of Government and the PM.
I might add that while we don’t directly elect the PM, at election time there’s no doubt in any voters’ mind who they are indirectly voting for as Head of Government. For example, I may not care for my local Federal Member of Parliament, but I will nonetheless vote for him because I want a particular candidate as the leader of this nation.