You can basically think of corporations as being like religions. Religions provide +1 happy and +1 culture to each city they're in, and all they require is that people be there. You can found them by being the first to discover a tech, and spread them with missionaries. If you have the holy site for a religion, then it generates money for each city that has the religion.
Corporations give a variety of effects to each city they're in, but they require certain resources to give those effects (which depends on the corp). You can found them by researching a tech and then spending a Great Person, and spread them with corporate executives. You automatically get the headquarters of the corporation when you found it, and it gets money for each city that has the corporation.
One other quirk about corps is that they can provide important strategic resources -- specifically, the Ethanol corp can give you access to oil, and the Aluminum corp can give you access to aluminum. You need the appropriate resources to feed into the corp to get that access, though. Still, they can be good ways to get those resources if you don't have any other access. Ethanol in particular as most of the late-game military units require oil.
Corporations give a variety of effects to each city they're in, but they require certain resources to give those effects (which depends on the corp). You can found them by researching a tech and then spending a Great Person, and spread them with corporate executives. You automatically get the headquarters of the corporation when you found it, and it gets money for each city that has the corporation.
One other quirk about corps is that they can provide important strategic resources -- specifically, the Ethanol corp can give you access to oil, and the Aluminum corp can give you access to aluminum. You need the appropriate resources to feed into the corp to get that access, though. Still, they can be good ways to get those resources if you don't have any other access. Ethanol in particular as most of the late-game military units require oil.