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 has complete and utter control over all of his subjects and people, and may also be commonly known as a dictatorship or totalitarianism. 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.
Monarchy
A monarchy is (obviously) a nation that is ruled by either a king, queen, or both a king and queen. Many of the troublesome aspects of a despotism no longer exist in a monarchy, and the economy is slightly boosted. The monarchy is a great mid-age government to play with.
Communism
A communism, also known as a socialism, is an ideal where everything and everybody is equal; nobody is more successful than each other. In a communism, all cities are equal; they are producing shields and commerce at the same amount, regardless of what resources appear around them. Communist governments do not receive the economic boost that democracies do. However, communisms would be the ideal modern government for those who love going to war.
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.
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.