[U][SIZE="3"][B]Hereditary Rule[/B][/SIZE][/U]

Required Tech: Monarchy
Effect: +1 per military unit stationed in a city
Use: For happiness till other civics are available in all game formats.

Hereditary Rule is the only civic you have available for a long time (till you get constitution) and is usually only used in that time-frame. The other government civics that get available later on, are a lot stronger, so you switch to them.
How to use Hereditary Rule is pretty obvious. Put as many units in your cities as you need. If you have to build military anyway, use that, otherwise get warriors. You can even deliberately not connect a city to resources, so you can build warriors in that city, while you have better units available in other cities. This way Hereditary Rule basically provides you with unlimited happiness at the cost of having to produce the units and having to pay maintenance for them.

[B][U][SIZE="3"]Representation[/SIZE][/U][/B]

Required Tech: Constitution
Effect: +3 per specialist; +3 in X (dependant to map-size) cities.
Use: Speeding up science in Ironman/ffa, and indu/modern teamers.

You use Representation whenever you gain a considerable amount of your science with specialists. Since you will be working Merchants or Scientists in most cases, Representation will double your science for those. This is most important for specialist economies in Ironman/ffa, but also when having to tech in indu/modern. Representation is usually the main reason for building pyramids in ffa/Ironman, so you can use Representation early, which provides a huge boost when playing a specialist economy.
The happiness is useful in Ironman/ffa as well, since its available in exactly those cities that have problems anyway. Because of this you can usually count it as three happiness per city, especially in early stages with Representation from Pyramids. In indu/modern you just work as much culture as you need anyway, because your base commerce is very low, so it doesn´t matter. When going for cottages in indu however, it might be a factor.

[B][U][SIZE="3"]Police State[/SIZE][/U][/B]

Required Tech: Fascism
Effect: +25% military unit production;-50% war
Use: Increasing production in modern/future teamers. Decreasing war weariness in non-always war games.

The most common use for Police State is increasing your production in modern and future era teamers, where you solely produce units.
You can also use Police State for unit-production in ironman/ffa of course, but usually you either have Universal Suffrage because it provides more production, or have Representation to keep up your tech. The point in those games is, that with Police State, Mt. Rushmore and Jails you can decrease war weariness to 0%. If you fight a long late-game war in non-always war games you might be forced to do so.

[B][U][SIZE="3"]Universal Suffrage[/SIZE][/U][/B]

Required Tech: Democracy
Effect: +1 from Town; Can spend money to finish production in a city.
Use: Buying units at the beginning of indu/modern/future teamers and in late game ffa/ironman wars; Additional production with cottage economies in any game type.

In MP teamers Universal Suffrage is used during the first turns of later era teamers to spend your initial money on buying workers/units to speed up your start.
Another use is producing units by excessive buying in later era teamers in Ironman/ffa. With a fully developed economy you can buy several units per turn, when having turned down your science. If you fight hard wars there, you might want to do that.
The best known and most powerful use of Universal Suffrage is for increasing production with cottage economies in ffa/Ironman games. With your citizens working about 80% cottages, this usually at least triples your production. With a cottage economy this is a must have in late-game. (You might be going Representation with Sushi+Mining inc. but usually even then getting building faster and then working science is better). When running Universal Suffrage with a cottage economy you will also get huge production bonuses from golden ages. While usually production tiles provide at least 3 hammers, but you work few of them, not a golden ages provides an additional hammer on close to all your tiles, nearly doubling your production.