Raw milk is simply milk as collected from the dairy animal which has not undergone pasteurization, or heating intended to reduce or destroy bacteria.
Milk as it leaves the udder (or in the case of human mothers, the breasts) is full of nutrients: fat, protein, carbohydrates (lactose), minerals, enzymes, antibodies and more. It is, at a minimum, the perfect and complete food for a baby if the same species to eat and to thrive upon. It also normally contains at least trace amounts of bacteria, which arrive on the outside of the teat and migrate into the teat canal. The vast majority of the time these bacteria are beneficial, are LAB (lactic acid bacteria), and are essentially harmless to us or are outright probiotic. Milk is almost the perfect environment for bacteria, just as it is almost the perfect food for human beings, and they thrive in it especially at body or near-body temperatures.
The issue is that there are various pathogenic bacteria which also thrive in it, and which will make us sick if we consume milk contaminated with them.
It is essential to collect the milk as cleanly as possible, using well-established techniques to minimize the likelihood of contamination. There is no 100% guarantee that milk collected cleanly will be pathogen-free, but combined with regular testing the risk is very minimal.
Pasteurization of milk is an attempt to eliminate pathogenic bacteria. It generally accomplishes that, but it changes the milk in the process in some ways that are not very beneficial. It's a classic trade off of safety vs quality, and it's not always obvious which milk to choose for a given situation.
In some regions it is very difficult to legally buy and use raw milk from ruminants. There is a long and somewhat complicated history to why these laws and regulations were put in place. The pendulum is swinging back, and in many countries and many states raw milk is becoming legally available to local consumers, and for farms to sell raw milk or dairy products derived from it without ever undergoing pasteurization.
When milk is heated, a few components within it are chemically changed. In particular, free calcium ions which are dissolved within the milk liquid will begin to precipitate out, leaving the pasteurized milk lower in calcium. (This is one reason why vitamin D / cholecalciferol is added to pasteurized milk in the grocery store, to restore the calcium levels somewhat.) This affects cheese making in that calcium must be added back, usually as calcium chloride, so the milk will still form a good curd. Raw milk generally does not have this problem, although in a few cases some added calcium chloride can benefit raw milk as well.
The heat of pasteurization also will denature some proteins and enzymes and destroy antibodies, leading to a less benefits when consumed. As there are different heat vs time pasteurization methods, some are more destructive than others with respect to denaturing of some components of the milk. Raw milk maximizes these components naturally.
On the flip side, raw milk retains its endemic cultures which find their way into the teat canal of the dairy animal, LAB or otherwise. If the milk is not cleanly collected, that may mean a higher count of pathogenic bacteria may be present, which can be dangerous especially to babies, those with weak immune systems, and the elderly. But when the milk is clean and well-collected the endemic bacteria are generally probiotic and helpful, not harmful. Clabber is the result of letting the endemic cultures work on the milk; raw milk rarely “goes bad” and becomes dangerous to drink when left out like pasteurized milk does.
It should be noted that pasteurized milk does not eliminate all bacteria, much less pathogenic bacteria. It can drastically reduce their population numbers, leading to greater safety from pathogenic infections, but there are still many documented outbreaks (and deaths) among consumers of properly refrigerated pasteurized milk. Both types benefit immensely from careful, clean collection and proper refrigeration.
Any cheese may be made with raw milk. For some cheeses, it doesn't make sense to use raw milk unless you have an abundance of it, because the milk gets pasteurized in the process: paneer, ricotta, etc. But for most types raw milk can provide a more complex or traditional result.
Note that some cheese styles can amplify any problems with milk, such as washed-curd cheese like Gouda or washed rind types like Reblochon because the pH of the final cheese is higher, giving pathogenic or undesirable bacteria more opportunity to grow. If you find you are having issues, you'll want to double-check the cleanliness if your raw milk, or use a gentle pasteurization of it before making cheese.
Like raw milk, any time the word “raw” is associated with other dairy products it indicates that it was created from raw milk and never underwent pasteurization. This includes the usual methods of separating cream, churning out butter from cream to get butter and buttermilk, and even ice cream.
Typically yogurt isn't available as a raw milk product because the milk is heated to 185 F and then cooled to 110 F and then cultured, which pasteurized the milk (this is done to deliberately denature the proteins and make a thicker yogurt). Euro-style yogurts may not be cooked like that and have a looser consistency, in which case they could be a raw-milk yogurt.
Weston A. Price Foundation site on raw milk benefits and more.
Farm to Consumer Legal Defense Fund information on raw milk and associated laws, plus more any ruminants, herdshares, and more.