Corporations have the potential to be a very powerful addition to your empire, potentially so powerful that they can turn a loss into a win. This article lays out some general rules about using corporations, and outlines some corporation strategies.
(Almost) Absolutes[]
There are a couple of things that are 99% guaranteed to be beneficial with regard to corporations. Outside of some very specific circumstances and strategies (which will be covered), you'll always want to do the following.
Choosing your path[]
Do not let the resources currently available to you be the main factor in your choice of what corporations to pursue; use the entire map to make that determination.
Most AIs will sell you extra resources quite cheaply once you have a corporation that uses them, so check each AI to see if it is friendly enough to trade you resources, and how many of the ones you want most (seafood, 
Rice, metals) it has. In general, AIs seem to charge 3 or 4 
Gold/turn for non-strategic resources, although the first one you buy from a given civ will be a bit more. Strategic resources (
Iron, 
Copper, etc.) go for considerably more — 12 Gold/turn or so. But even at that price, they are well worth buying if you have Mining Inc. spread to many cities.
There is also nothing forbidding you from invading other civs to take the resources that will allow your corporations (and, by extension, your entire empire) to thrive.
Found Sid's Sushi and Mining Inc. and spam them[]
Sid's Sushi Co. and Mining Inc. are the two most powerful corporations, Sid's for 
Food, and Mining for 
Hammers. Found these two as soon as you can, and spread them to most of your cities.
Why are these two corps so great? Well, Mining Inc. uses many resources and produces only 
Hammers; it is great in every city you own. Every city can use more Hammers, without exception. The only cities you should not spread Mining to are your 
Culture cities (if you are going to try a cultural win), and to your Wall Street city (if you plan to found lesser corps for spread to the AI).
Sid's is only slightly less amazing. It also takes many resources, including seafood resources which tend to be numerous on most types of maps. It produces Food and Culture, so you'll want it in any city which:
- Needs
Culture (to push borders, or if going for cultural win). - Needs more Food to work low-food tiles (i.e., plains, hills, tundra, even ice).
- Can use more Food to downshift tiles from Food production tile improvements (Farms, Windmills) to Commerce or Hammer improvements.
- Can use more Food to run specialists and make them pay via Great Person generation.
On some maps, seafood is not plentiful or even almost absent (i.e., Great Plains). On this sort of map, Sid's is not a good idea; you'll want to wait for Cereal Mills. Cereal Mills produces only Food, and is efficient. So it's a decent corp.
Headquarters[]
Found all corporations in a city with as many Gold multiplier buildings as possible. That is, make sure the city or cities you use have a Market, a Grocer, and a Bank. This gets +100% Gold, giving you +8 Gold/turn per city to help offset the high maintenance costs that corporations cause in the cities they are in. Found as many corps as possible in the city that is, or will become, your Wall Street city. It will get +200% Gold, +12 Gold/turn per city. Note that because corps compete, you cannot found them all in your Wall Street. Therefore, it is better to plan your corporate strategy far enough in advance to decide on whether or not to put Mining Inc. there, or elsewhere. If you plan to found one or more minor corps for spreading to AIs, then you should put Mining Inc somewhere other than Wall Street. If you plan to found only Mining Inc and Sid's Sushi, leaving the lesser corps to the AIs, then both should go into Wall Street.
Foreign spread[]
Restrict the better corporations (Sid's Sushi Co., Mining Inc., and sometimes Cereal Mills) to being your own private corporations, rather than spreading them out to other civilizations. Other civs get essentially the same benefits as you do from a corporation (although the founder always benefits more in getting the Gold in the Corp HQ city). You don't want them getting the better corps. Furthermore, when other civs get a corp that uses a particular resource, they will keep any spare resources of that type for themselves. Then they won't sell them to you. And they will compete with you to buy resources from 3rd-party civs.
There are corps which are (or can be) safely spread to other civs. These are corps which do not yield a very strong benefit in some or most circumstances. Let's look at them, in order of their general utility, lowest to highest.
The corp best suited for foreign use is Civilized Jewelers Inc. Its main product is Culture, which is useless in most cities most of the time. CJI also produces a small amount of Gold (1 per resource on Standard-sized maps), but the additional maintenance cost for the corp will eat up most of that. CJ has only one downside for foreign use: it uses 
Silver and 
Gold, and will thus tend to decrease your benefit from Mining Inc. (because AIs won't sell you their extras). In addition, you will of course want to avoid giving it to any civ that is going for a cultural win, at least if you are not strong enough to stop it by force.
Aluminum Co. is another corp well-suited for foreign use. It uses only 
Coal, which is usually in short enough supply that foreigners won't get much benefit, nor will you lose too many resources for Mining Inc. It does return 3 
Science per resource (on Standard-sized map), so, it will benefit any civ somewhat. Also, Aluminum Co. supplies 
Aluminum, which may matter in a Space Race if a civ does not already have it.
Standard Ethanol is similar to Aluminum Co. It uses Rice, so giving it to AIs may reduce available Rice for Sid's Sushi. It returns 2 Science per resource (on Standard-sized maps), so its benefits are fairly limited. And it supplies 
Oil, which is useless to most civs (natural Oil is not rare), but may be very important to deny it to a few. On the other hand, it is more powerful than Aluminum Co. in that it uses three resources ( Rice, 
Corn, and 
Sugar), not just one.
Finally, Creative Constructions is another relatively weak corp for most civs/cities, in that it creates a lot of Culture. However, it does still create 0.5 Hammers per resource. So it is far from innocuous when an AI has it. CC also competes with Mining Inc., although the conflict is over Copper and Iron, both of which are strategic metals that the AI won't sell cheaply.
Strategies[]
There are many strong strategies if you are going to build corporations at all. One is to build only the two best corps yourself, and keep them private. Another is to found many corps (four, at least, and maybe more), keeping the two best to yourself, and using the others as money generators by spreading them to foreign civs.
Sid's and Mining only route[]
If you are going to found and use corps at all, then these are the two you want. Found them both in your Wall Street, buy up resources from AIs, and have fun with your super-size cites with great Production.
Foreign earnings route (1)[]
In this strategy, your plan is to make money from spreading one or two corporations to other civilizations, so you plan to found four in all: Sid's and Mining for private use, and Civilized Jewelers and Aluminum Co. for foreign use. Sid's can coexist with both of the two lesser corps, so all three of them should be founded in your Wall Street. Mining Inc. competes with both lesser corps and must be founded elsewhere.
When should you do this rather than Sid's and Mining only? Basically, when going for any type of win other than military wins. Military wins end the game early, tech-wise, and (assuming you are the one winning), with you controlling most of the world. In these circumstances, you won't be getting much or anything from corps that can only be founded late in the game, and spreading corps yourself to a huge empire is a hassle enough even for two corps.
Sometimes you'll generate a more-or-less unwanted Great Artist later in the game after going the "Sid's and Mining only" route. One good use for him is a minor variant of this strategy: you found Civ Jewelers with him and spread it. The money is not as good (CJ not being in Wall Street), but otherwise it is a good use of a GP.
Foreign earnings route (2)[]
This strategy is like the previous one, but the plan is to make even more money from foreign corps. The plan is to found Standard Ethanol (and possibly also Creative Constructions), in your Wall Street when you can. This means you'll have to found both Sid's and Mining in some other city.
This strategy leans even harder on your ability to generate Great People, so you should try it only if you know you can generate the 5+ GPs you need. And it pays out even later, so it is most appropriate when you know you cannot win via the U.N. or warfare, but will winning via a Space Race or even a Time Victory.