There are six kinds of governments--anarchies, despotisms, monarchies, communisms, republics, and democracies.
Anarchy
An anarchy is basically a mass state of chaos in a nation. In Civilization, nothing can be created by your cities, and the economy infrastructure collapses, leaving you either in debt or losing gold. Many cities also fall into civil disorder. However, a state of anarchy is necessary for a change in government, and is only temporary. Anachies usually only last 2 to 4 turns.
Despotism
A despotism is where a ruler dominates the civilization, but not to the established extent of a Monarchy (meaning that the ruler could be anything from an opportunistic general to something else entirely). A nation begins with this government at the start of the game. Gold is easily produced, but not in large quantities. It costs the lives of citizens to rush construction projects, which is an annoying aspect of a despotist government. Corruption is a massive issue as well.
Monarchy
A monarchy is (obviously) a nation that is ruled by either a king, queen, or both a king and queen. Corruption is still a problem, but is greatly reduced from Despotism. Just as importantly, Unit Rush Builds can now be payed for with gold rather than the lives of your workers.
Monarchy is an excellent early-to-mid-game government if your civilization is surrounded by other civilizations that may or may not turn on you and attack, effectively mandating permanent base garrisons. It allows a flat 4 free units in terms of support before the units begin to cost gold each, and allows up to three units to serve as garrison units (and police the city). It avoids Republic's and Democracy's high unit support costs, as well as issues with war weariness.
Communism
Communism, at least in the context of this game, refers to government ownership of the major means of production. Because the economy can be re-oriented to focus heavily on war-time production, a Communist civilization gets major production and war bonuses (and does not need to fear War Weariness to the degree that a Democracy might).
However, Communism is terrible for a civilization's economy, grossly reducing income. That, combined with Communism's poor science rating, results in drastically slower research.
Republic
The prelude to a democracy, a republic is a government where the people of the country chooses what the country does and where the country is headed economically, politically, and with war. The economic infrastructure is boosted in a republic, and more than a monarchy, as each person has all of their personal freedoms. However, should a country go to war, civil disorder might strike the nation's cities due to the war-weary protestors of your nation.
Republic has several advantages mentioned above, as well as one big early-to-middle game drawback: namely, that you get no units free of a "support" cost (meaning cold paid out of the treasury per turn to keep them in the field). That's not much of a problem in middle-to-late games, where the total amount required is (hopefully) dwarved by the amount of money coming into your treasury. In early games, though, that amount of gold can seriously hurt your bottom line.
Democracy
A democratic government is literally just a republic with one difference: the people elect other people to decide what they where the country will be headed. In a democracy, the economic infrastructure is amplified once again, and many players who become a democracy should be earning hundreds of gold per turn if they handle their nation correctly. A democratic nation should be the ideal choice for those who wish to build.
The main drawback with Democracy is that War Weariness is a problem. Never begin wars with Democracies, at all costs - instead, provoke an enemy into declaring war on you. Even then, a Democracy at war long enough will eventually begin to suffer War Weariness issues.