Parliament
A parliament is a national or regional legislature. More specifically, the term "Parliament" (without the definite article, and with a capital "P") is frequently used to denote the legislature of the United Kingdom, which is composed of the Queen, the House of Lords and the House of Commons.
The "Parl-" root means "to speak," as in parley.
Organization
Some parliaments have proportional representation--that is, the party that wins the most votes in an election supplies an amount of representatives proportional to how much of the vote they got. This is different from American "winner take all" elections, in which whoever wins the majority in a particular district supplies the only representative. Winner take all elections are more likely to produce a two party government, but proportional systems allow third parties a chance to win some representation if they can win a small amount of the national vote without winning a majority of any districts.
Heads of government in parliamentary systems are the leaders of the party in power. This person is referred to as the "prime minister." This is in contrast to heads of government in presidential systems, who are not members of the legislature and who are not elected out of the legislature. The prime minister remains in power not for a fixed term but for as long as he has the "confidence" of Parliament--that is, as long as he enjoys their support. The Prime Minister is removed upon losing a vote of no confidence, and some resign after losing a major vote on an issue they have championed (as Italian Prime Minister Romano Prodi did in February 2007).
Rules of order
Parliaments and other governing bodies have "rules of order" which govern who may speak, and how votes are taken and counted. These can be very complicated.
To see why these rules are needed, consider this possibility. Suppose a vote is taken in the U. S. House of Representatives when only 300 of its 435 members are there. Suppose 200 people vote for the motion and 100 people vote against it. Should the motion pass? A majority of those who were present voted for it, but the 200 people who voted for it are less than half of the entire membership. There's no way to tell how the 135 absent members would have voted; if all of them had voted against the motion, it would have been defeated.
In this case, the rule that applies is called the quorum rule, and it says that as long as there are at least 218 members present, votes can be taken and the majority of those present prevails. So, a clever representative may try to arrange for votes to be taken at a time when he knows that more of his supporters and fewer of his opponents are present.
"Rules of order" are often called parliamentary procedure. Many organizations adopt rules of order which are broadly similar to each other. A popular reference book called "Robert's Rules of Order" gives a detailed outline of parliamentary procedure. In a small organization, like the governing body of a club or a town government, many members may not be completely familiar with all the rules. There may be a member called a "parliamentarian" whose job is to know the rules and act as a sort of referee for enforcing them.