Republics are great governments for early growth and trade, and early-game republics often facilitate growth. The drawback to republics is that no unit comes free of support, and each ground or sea unit with attacking power beyond the first (and each air unit) that is outside of a city or a fortress up to three squares from a friendly city will make a citizen unhappy. There is also no military police, making improvements such as temples and cathedrals more important. Settlers and Engineers also require more food for support compared to other governments, requiring two.
Republics also enjoy a trade bonus, as well as a maximum rate of 80%. A republican city will grow by one while celebrating "We Love the King Day" as long as it has sufficient food.
Democracy is arguably the best form of government due to its numerous abilities. There is no restriction on maximum rate, and cities generate extra trade. As well, units are immune to bribery and it costs more for other civs to bribe a city under a democracy. There is no loss of shields or trade due to corruption and waste, and courthouses make one content citizen happy. If a city in democracy celebrates "We Love the King Day", the city will grow by one citizen each turn as long as it has sufficient food.
However, these benefits come at a price: there is no free support, and so all units must be supported with a shield from the unit's home city, and each unit (with attack power) that is not in a city or a fortress within three squares of a friendly city will make two people in the unit's home city unhappy. To make things worse, air will always make two people from the unit's home city unhappy. This makes improvements such as temples and cathedrals that much more important, as well as having the Shakespeare's Theater wonder in a city with high shield output. Democracies are also very fragile, as all civil disorders must be taken care of immediately or the citizens revolt. Settlers and Engineers also require two food from its home city to support.
Democracies also suffer from its relative inability to declare war. The senate will prevent almost every attempt at warmongering. Thus, the United Nations wonder is a valuable asset to any democracy, as it reduces the senate's influence to a more tolerable level.