I am sure a more experienced (and better) player will add to/correct this post. The merits of each government vary depending on the situation you find yourself in. I'll simply list each one and give an idea of when you would/would not use them.
Despotism, you start out in this.
You can use your military to keep people happy and therefore working which in turn means you make stuff.
Each city can support a number of troops depending on its size. Each tile that the city is 'working' will not be used to its full efficiency.
You can hurry building of stuff by whipping the population of a city, this makes the item instantly but loses some population from the city and makes the rest unhappy.
You can go to war at will.
Corruption will be very bad and hamper any city even relativly short distances from your capital
Most players will try to get out of despotism ASAP.
Monarchy (i don't often use Monarchy)
Generally used by the warhungry
Cities support a number of troops depending on size
Troops can be used to keep people happy
Tiles use is better than despotism
You can goto war for a long time with minimal (if any?) unhappiness.
Corruption will effect your output, both financially and productivly, hence most people use this simply to wage war.
Republic (my favorite)
Troops can not be used to make people happy
All troops are paid for from the treasury - be careful switching to republic if you have a massive military as you may not be able to afford it.
Corruption is not a major problem, but can still hamper far flung cities. As a consequence output of money, shields and research is great.
I think republic is by far the best one to use for most situations, you can wage war although unhappiness will kick in eventually. Research and income are really good and your production will be able to churn out the war machine/improvements.
Communism (probably the least used)
Similar to Monarchy in it allows you to wage war at will with not much unhappiness.
A city can support a number of troops depending on size, this is more then monarchy.
Troops make people happy
Corruption is evened out over your whole empire so far flung cities will experience very little corruption than they would under a different government type BUT your core cities, which are your most productive, will also suffer more corruption than they probably would. This can really hurt your production, research and finances but is really good for going to war.
Democracy A more advanced republic
Corruption is lowered compared to republic
Workers work harder and faster
Otherwise pretty much like republic except war weariness will kick in very quickly and simply destory your output in a long war.
I think the general consensus is Republic is probably the best 'jack of all trades' government and a lot of players will try to get it as quickly as possible. Regardless though all government choices are much better than despotism so you should get into Monarchy or Republic ASAP and dont wait for Democracy or Communism. Also the anarchy period between switching govs will hamper your empire, unless your playing a religous civ stick with Republic or Monarchy. The loss of output going from Republic to Democracy during the anarchy will take a long time (100+ turns) to recover. If your a religous civ change at will. Hope that helps some.
