Mathilda:
First off welcome to CFC
Let me describe all the governments:
Anarchy: The worst there is with 100% corruption, no production, food penalties, slower workers and no rushing of projects
Despotism: Just plain bad in most areas with food & shield penalties, high corruption and whip rushing of projects
Feudalism: Only good for lots of small cities has corruption though not as bad as despotism allows for a reasonably large military, but as cities grow costs jump for support and can only whip rush.
Monarchy: Best choice for the early warmonger, similar corruption as Feudalism, yet unit support grows instead of shrinks as cities grow and can cash rush projects.
Democracy: The best for a builder. No unit support is only drawback, but workers are faster, corruption is minimal and commerce ($$) has a bonus. A late Industrial Age powerhouse government. In the late Industrial Age, this is the place to be unless there are war clouds on the horizon. If you have to fight make the wars quick and a blood free as possible. Try to get lots of allies and have them do the warring for you while you research to the stars.
Republic: All around the best choice for a government. More corruption than democracy, but less than Monarchy. Get the same commerce boost as Democracy but no worker bonus. You have to watch you military size as units can get expensive. You can wage longer and bloodier wars (particularly if you can get Universal Suffrage).
Each government has it's strengths and weaknesses and each has a niche that is the right place for it. If you happen to be religious, then switching between them is the best option. Democracy for peace time, moving to Republic if war breaks out then to Monarchy/Communism if the war drags on. However, if you aren't religious then the anarchy periods can be real killers on science & production, so it's best to get to one tech and stay there and Republic is the best early tech so that's where most of the players go for.
I should note that I've never gone to Fasicsm so I don't know much about it and my turns in Communism are very few so I'm not a fair evaluator of that type either.
There is an excellent write up by SirPleib comparing Monarchy to Republic, if I can find it, I'll post the link here:
Edit: Looks like I typed way to slow to get this in first and Arathorn said it better than I.