A parliamentary system is a form of government where the executive branch -- responsible for the operation of the state bureaucracy -- and legislative branch -- responsible for enacting the laws of the land -- are more intertwined, as opposed to a US-style representative republic, in which the two branches are distinctly separate. Parliamentary governments, which exist in Canada, Australia, the UK and many other countries, have a number of features that some may see as advantages or disadvantages, depending on their point of view.
Legislation Passes Quickly
In a parliamentary system, it is easier to pass legislation as the executive is accountable to the legislative body. This is in contrast to a presidential system where the executive is chosen in a separate election and has the power to veto legislation. Whereas the executive in a parliamentary system is typically from the majority party, in a presidential system the executive is often from a different political party than the legislative body, resulting in stalemates. Some might say it is a good thing that parliamentary systems can pass legislation more quickly so they get more done, but others might say such a system lacks necessary checks and balances.
Easy to Remove the Executive
Because the executive is accountable to the legislature in a parliamentary system, it's much easier to remove the executive than in a presidential one. If a prime minister loses the support of parliament, it can force out the executive without great upheaval. The parliament did this to former prime minister Margaret Thatcher in 1990. In a presidential system, the executive may only be removed in the case of criminal activity. Some might say this is an advantage of a parliamentary system as a president who is unpopular but difficult to remove will become a "lame duck" or powerless figure. However, some might say this is a disadvantage as it eliminates the check an executive provides against the whims of the legislature.
Some criticise the parliamentary system for being too democratic, pushing aside the needs of the minority in favour of the majority. Whereas in a presidential system a legislative body is held in check by an executive and a judicial branch of government, in a parliamentary system the legislature is the most influential part of the government. However, some argue that the presidential system is flawed because voters democratically elect both the legislative and executive branches anyway, and thus both can claim a democratic mandate, whereas voters choose a single legislative and executive body in a parliamentary system.
Power Spread Out
Power is more divided in a parliamentary system, spread out among a number of leaders in parliament, including the prime minister, the majority party leader and the whip, to name a few. A prime minister does not have as much power as a president, and thus political ideas are more important than individual people. In a parliamentary system, a shift in power can take place without an election.