Manufactories are a kind of city upgrade. Manufactories are very expensive, but they are among the best investments in the game. Their primary effect is automatically investing in research or stability, but they also yield a small production income.
Perhaps the primary effect of manufactories is to automatically invest for you in one of the tech fields, or in stability. The investment is 5 ducats per month -- 60 per year. A second major effect is in raising your trade or production efficiency, lowering your stability cost by a small percentage, or in raising a military support limit. Both effects vary with the manufactory type; see below.
Note that the effect on trade efficiency or production efficiency can create a tremendous amount of income, depending on how much your empire trades or produces. For example, each refinery adds +1% to your trade efficiency. In addition to making your merchants slightly stickier, if you've gotten a substantial fraction of world trade, this can be a very great amount. Total world trade later in the game will exceed 20000 ducats per year, with a human player often getting 40% of it. So, it's worth 8000 ducats times your trade efficiency. A 1% boost is thus worth 80 ducats per year. Trade taxes from the provinces you own are also affected by trade effeciency; you get even more additional income from a boost in trade efficiency.
Manufactories also create a small production income. This income will be 6 ducats per year, or 12 ducats per year if the province is the right type. The extra 6 ducats you get in a province of the right type is called the "production bonus".
Types of Manufactories
There are five types of manufactories.
Fine arts academies
Fine arts academies (abbreviated FAA) are the first manufactories to become available. They get the production bonus if in a capital province. FAAs lower a country's stability costs slightly (by 1% each to a maximum of 50% reduction), and they generate direct investment in stability. Most players don't build them, or they build just one, because stability is usually either high all the time (for small countries), or else it is not invested in.
Refineries (also known as breweries) get the production bonus in a sugar or wine province. In addition, they raise a country's trade efficiency and generate direct investment into trade tech. Most players consider refineries the best investment in the game: they're available early, and the trade efficiency effect benefits any country with a strong presence in world trade, be it large or small.
Goods manufactories get the production bonus if built in a cloth, cotton, or tobacco province. In addition, they raise a country's production efficiency and generate direct investment into infrastructure tech. Goods manufactories can be the best investment for very large empires, but they won't be as beneficial as refineries for physically small trading empires (which don't produce much). Goods manufactories are also only buildable much later in the game.
Naval equipment manufactories get the production bonus if built in a fish or naval goods province. They also increase a country's naval support limit, and generate direct investment into naval technology.
Every time an enemy army (including rebels) is present at the end of a month in a province that has a manufactory, there is a small chance that the manufactory will be burnt (2% chance from EU2 readme). You are not notified when this happens.
Manufactories can also be lost by event. There is one random event, Devastating Fire, which causes the loss of a randomly chosen manufactory. However it only happens to countries which own at least 30 provinces. Some major powers have scripted events which cause loss of manufactories, for example the Spanish Bankruptcy Events.
If you are innovative enough (>=7 or >=9 for 2x) you are eligible for a random event called "unexpected invention" that will place a random manufactory in a random province. Free manufactories are great and can save you thousands of ducats when the costs are high. It will probably be the wrong type of manufactory for the province, but it is even worse when it hits a province that already has a manufactory, changing it to the wrong type for a net loss. Some lucky countries gain a manufactory through a scripted event, and then it is usually of the right type.
Cost of manufactories
There isn't an exact formula for the cost of manufactories, there is an approximation. The cost of a manufactury is mainly dependenant on the amount of other factories: (Base cost) + 250 * (# of finished manufactories of the same type) + 150 * (# of finished manufactories of a different type) + 150 * (# of unfinished manufactories). The difficulty level affects the base cost. The ADM rating of the monarch and time also have some influence.
Where do I get the money? you say. Manufactories are well worth minting money to get. Go ahead and seriously mint to get them; just make sure you don't lose the money in peace deals while saving. (The enemy knows how much you've got when he offers peace.) The sooner you build them the more time you have to benefit from them. You should stop building them around 1750, when they are probably very expensive and will not give you a return.