Also here is a nice breakdown of the main types of world governments. There is a difference between a Republic and a Democracy.
Most nations in Europe I would consider Social Democracies. The US for example is a Constitutional Republic, and Canada is a Constitutional Monarchy. China is a mixed bag of a Capitalist/Socialist Communist Dictatorship.
Nations like N. Korea and Cuba are Communist Dictatorships. Other nations like Venezuela is a Democratic Dictatorship. I know it sounds like a oxymoron, in reality it isn't though. You still have "rights" and can "vote" but it's monitored by the government to make sure you are voting for the dictator.
A very classic categorization of governments stems from ancient Greek Aristotle: He distinguished 6 types of government.
Either one person rules (when he is a good, well meaning leader, then it's #1 monarchy; when the leader exploits his position, it's a #2 tyranny).
Or few people rule (when they govern well, it's #3 aristocracy, when they exploit the people, it's an #4 oligarchy).
And finally, when all people rule (when they do it well, it's #5 a polity, when it becomes a horrible mob rule, it's #6 democracy).
The ideal type of government, he believed, would be a constitution that contains elements of all three types. And indeed, our Western democratic republics of today are such "mixed constitutions". For example, the USA: One rules (President), the few rule (members of Senate and parliaments), all rule (the entire people elects Congressmen and President).
Although often referred to as "democracies", modern Western states today are what philosophers of the Enlightenment, such as Montesquieu for example, called republic: Unlike in a democracy, where indeed the entire people takes *all* decisions, the people only elects representatives in a republic, who then govern in their name. Also, there is a seperation between the branches (executive, legislative and judicative) in a republic, unlike in a genuine democracy, and a constitution that guarantees the protection of basic civil rights.
These kind of republics are also referred to as representative democracy, or pluralist democracy. But of course, the constitutions of different modern republics are very different from each other -- there are presidential democracies (i.e. USA), parliamentary democracies (i.e. Britain, Germany and many others) or semi-presidential democracies (i.e. France, Poland) -- depending which office/organ plays the more important role, president or parliament.
In the past, there existed many people's republics, which used to be communist one-party dictatorships with only insufficient protection of civil rights and nationalized, state-run economies.
When the formal head of state is a monarch, yet the parliament plays a central role, this system is called parliamentary monarchy (i.e. United Kingdom, Spain). When a monarch rules without public control and does not even respect a constitution, it's called absolute monarchy (i.e. France before the Revolution 1789, Saudi-Arabia today).
Of course, there are many other types of government.